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1. rem SALFORD FTN 7 Linker rem rem SLINK 1 OBJ FILE 951 LINK 2 AUX rem pause rem PRIN rem WATCOM F77 Linker rem rem SET WATCOM C FORTRANWYATCOM rem BINBWVFL386 1 0BJ FIFLINK 2 AUX FE 1 L WIN385 rem BINBWVBIND 1 n s binwAVVIN3S86 EXT rem DEL 1 REX rem MS Fortran PowerStation Linker rem SET LIB LIB LIB rem BINALINIK NODEFAULTLIB QVVIN LIBF NOLOGO OUT 1 EXE 1 0BJ LINK 2 AUX Lu rem Compaq Visual Fortran Linker rem call c program files microsoft visual studio d98 bin dfvars bat df link out 951 exe 951 0BJ LINK 2 AUX lapackidvf_lapack lib rem pause KEM En rem ABSOFT Fortran Linker ror Figure 8 15 Linker Options 19 Edit PDEFIT INC E xj LWAL LENGTH OF FIRST REAL WORKING ARRAY FOR OPTIMIZATION ROUTINES MUST BE SUFFICIENTLY BIG LWA2 LENGTH OF SECOND REAL WORKING ARRAY FOR SS OPTIMIZATION ROUTINES MUST BE SUFFICIENTLY BIG ave ss LWA3 LENGTH OF THIRD REAL WORKING ARRAY USED ONLY FOR EXECUTING ODE SOLVERS SIZE DEPENDS ON ODE METHOD BANDWIDTH AND DISCRETIZATION ACCURACY LKWA LENGTH OF INTEGER WORKING ARRAY FOR OPTIMIZATION Compile Lu ROUTINES MUST BE SUFFICIENTIY BIG LWLOG LENGTH OF LOGICAL WORKING ARRAY FOR OPTIMIZATION ROUTINES MUST BE SUFFICIENTIY BIG LFLUXP LENGTH OF WORKING ARRAY FLUXPD FOR STORING DERIVATIVES OF FLUX FUNCTIONS MUST BE NOT Link SMALLER THAN 3
2. C C Problem LKIN_X e e l J C VARIABLE ki k2 D t C FUNCTION hi hi D EXP k1 t C FUNCTION h2 h2 ki D k1 k2 EXP k2 t EXP ki t C END C 6 2 Laplace Transformations The input of variables and Laplace functions is very similar to the input of explicit model functions Variables are 1 The first variable names are identifiers for the n independent parameters to be esti mated pi pn 2 If a concentration variable exists then a variable name must be added next that represents the concentration variable c 3 The last variable name identifies the independent variable s in the Laplace space that corresponds to the time variable t after back transformation for which measurements are available 4 Any other variables are not allowed to be declared Since constraints are not allowed the only functions that can be declared are r fitting criteria formulated as functions in the Laplace space any functions H p s c fork 1 r depending on p s and c Any other functions are not permitted These functions are then transformed back to the original variable space in the time variable t The constants n and r are defined in the database of EASY FIT 9 and must coincide with the corresponding numbers of variables and functions respectively Example To illustrate the usage of function input in th
3. 237 238 239 240 241 Horst R Pardalos P M eds 1995 Handbook of Global Optimization Kluwe Academic Publishers Dordrecht Boston London Hotchkiss S A M 1992 Skin as a xenobiotic metabolizing organ in Process in Drug Metabolism G G Gibson ed Taylor and Francis Ltd London 217 262 Houghton D D Kasahara A 1968 Nonlinear shallow flow over an isolated ridge Commu nications of Pure and Applied Mathematics Vol 21 1 23 Hughes W F Brighton J A 1991 Theory and Problems of Fluid Dynamics McGraw Hill New York Hull T E Enright W H Fellen B M Sedgwick A E 1972 Comparing numerical methods for ordinary differential equations SIAM Journal on Numerical Analysis Vol 9 603 637 Igler B Knabner P 1997 Structural identification of nonlinear coefficient functions in transport processes through porous media Preprint No 221 Dept of Applied Mathematics University of Erlangen 1997 Igler B Totsche K U Knabner P 1997 Unbiased identification of nonlinear sorption characteristics by soil column breakthrough experiments Preprint no 224 Dept of Applied Mathematics University of Erlangen Ihme F Flaxa V 1991 Intensivk hlung von Fein und Mittelstahl Stahl und Eisen Vol 112 75 81 Ingham J Dunn LJ Heinzle E Prenosil J E 1994 Chemical Engineering Dynamics VCH Weinheim Jacobson D H Mayne D Q 1970 Differential Dynamic Programming American Elsevier
4. DATE Plot output third line of information block e g date uu USER Plot fourth line of information block e g user name Karaca SE T AXIS Name for t axis time a6 4x 2i5 NPAR Number of parameters to be optimized must be at least one NBPV Number of variable break points i e the last NBPV variables are treated as break points where integration is restarted NRES Total number of constraints without bounds a NEQU Number of equality constraints a6 4x 2 20 4 RT RC Formatted input of NRES rows each containing two real numbers identifying the experimental time and concen tration parameters for which a constraint is to be sup plied The order is arbitrary but first the equality and subsequently the inequality constraints are to be de fined The data are rounded to the nearest actual time and concentration value 14 a6 4x i5 NODE Number of differential equations NODE can be zero if no differential equation is defined 15 a6 4x i5 NCONC Number of concentration values NCONC must be 1 or higher 16 a6 4x 215 NTIME Number of time points must be greater than 0 MPLOT Logarithmic scaling of x axis MPLOT 1 or not MPLOT 0 17 a6 4x i5 NMEAS Number of measurement sets i e of model func tions with respect to which measurements are supplied NMEAS is the dimension of the fitting function 18 a6 4x 15 NPLOT Number of plot points to be computed by additional model function ev
5. l which are to be treated as optimization variables in our optimum design problem 3 15 and which becomes min trace C w p q gi p q 0 j 1 Me g p q 0 f me lm s qeR weR 7 3 22 oe Wk 1 H T lt uk y ed ue with covariance matrix C w p q I w p q 1 depending now on additional weights I w p q F w p q F w p q F w p q V f w p q and finally f w p q wi h p q 1 wih p q t Note that for stability reasons a small lower bound 7 is introduced for the weights Corresponding partial derivatives of the objective function p w q trace C w p q subject to a weight zu are obtained from d o Ce w q trace ropa so Fes Fw Wk Ow 3 23 m Kop 7 id 101 40 35 30 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 Figure 3 10 State Function Cs t see 3 16 and from o o By Ab q Bu rfr p q d uni 3 24 Ow ODi i 1 np k 1 1 o i 1 np k 1 l confer also 3 17 Thus we get the weight derivatives more or less for free since the partial derivatives subject to the model parameters are known from the computation of the objective function Example 3 6 MICGROWX Y Z We consider again Example 3 4 see Banga et al 22 In addition to the model and design parameters also weights are to be computed at an equidis tant grid of 43 time values Thus the optimization problem 3 22 gets 86 addi
6. Next names of n variables belonging to coupled differential algebraic equations are defined v1 Une Where first the differential then the algebraic variables must be given If flux functions are to be inserted into the right hand side formulation of the PDAE then n identifiers for the fluxes and their spatial derivatives are given fi fn and Jis st ICH Then a name is to be defined for the space or spatial variable x The last name identifies the independent time variable t for which measurements are available Any other variables are not allowed to be declared Model functions are defined in the following format 1 If flux functions are to be used then nn functions HI u us v t a Du Uns Ed defining the flux are inserted one set for each integration area i 1 Ma They may depend on 2 t u uz and p Functions for the right hand side of partial differential equations FI tee P o 5 dt are defined next one set for each integration area i 1 ma Each function may depend on z t v U Ur Ure p and optionally also on the flux functions and their derivatives First the differential equations then the algebraic equations must be defined 13 10 Then corresponding initial values at time 0 must be set ui p x 1 1 Ma where first initial values for the differential and then for the algebraic equations must be declared They depend on x and p
7. 1 The first variable names identify the n independent parameters to be estimated pi zy Pre 2 The subsequent names specify the state variables of the partial differential equations Ui Um 3 In a similar way the names of the corresponding variables denoting the first and second spatial derivatives are to be declared in this order wiz Un and uis un 4 Next the names of n state variables belonging to coupled ordinary differential equa tions must be defined v1 Un iet 5 If flux functions are to be inserted into the right hand side formulation of the PDE then n identifiers for the flux and their spatial derivatives are to be given fi and J145 Jugo 6 Then a name is to be defined for the space or spatial variable z 7 The last name identifies the independent time variable t for which measurements are available 8 Any other variables are not allowed to be declared In a similar way we have rules for the sequence by which the model functions are defined 1 If flux functions are to be used then man functions PI u us x t Ta Dou diss d defining the flux must be inserted one set for each integration area 1 m They may depend on z t u u and p When evaluating the right hand side of model equations subsequently then the values of these flux functions and their derivatives are passed to the identifier names and corresponding derivative variables declare
8. 2 19 Y k12Y1 y1 0 D Yo ki2y1 haya y2 0 0 9 q residual pi p 4 0 212 107 0 10015 0 049823 5 0 105 1074 0 100120 0 049960 6 0 165 1075 0 100081 0 049963 7 0 104 107 0 100067 0 049953 8 0 101 1075 0 100063 0 049950 9 0 101 1075 0 100062 0 049949 10 0 101 107 0 100060 0 049948 0 462 1077 0 100054 0 050002 able 2 2 Final Residuals and Solution Vectors If Y and Y gt denote the Laplace transforms of y and ya respectively and if we exploit the linearity of the Laplace operator we get the system ep D kio Y gt sY kuch ka1Y2 Let Y p s D and Ya p s D be the solution of this system with p ez kaf i e D s ki k12D s k12 s ka Y p s D Ya p s D The parameters to be estimated are the transition coefficients ki and ka and three values are given for the initial dose D Experimental data are simulated in the following way We proceed from the time values given in Example 2 2 and p 0 1 0 05 7 compute exact solution values for D 50 100 150 and add a small random noise in the order of 1074 Then we execute the least squares code DFNLP with termination accuracy 10 starting from p 1 0 0 1 7 for different values of q The numerical results are listed in Table 2 2 where the last line contains the results obtained for the exact solution DFNLIP converges within 21 iterations in all cases We see that the residual is improved from q 4
9. 2 y 0510 20 30 do 50 60 70 80 Figure 3 7 State Functions n4 t ns t ne t 1 08st opt T t 04 02 Jj Ee u 00 10 20 30 40 50 60 70 80 Figure 3 8 Control Functions feed t and feed t 99 Table 3 8 Confidence Intervals for Urethane Problem before and after Experimental Design p initial final weights ken 1 14 14 70 7 44 Kan 56 45 11 53 2 73 ref 1 62 1 30 0 20 ka 859 13 471 0 00058 Eu 0 77 0 18 2 34 ES 1 24 1 95 0 95 Ens 1 38 0 84 0 14 dno 676 79 6 21 0 00026 500 450 400 AA 7 350r js 300 y 250 i 200 0 10 20 30 40 50 60 70 80 Figure 3 9 Temperature T t 100 3 4 Experimental Design with Weights The experimental design approach introduced in the previous section assumes that the time values are known in advance However there are very many situations where one would like to know in advance their approximate number and also their optimal locations to improve the confidence intervals of the parameters to be estimated and to reduce the number of time consuming or expensive experiments Our idea is to proceed from a given set of time values which could be large and dense and to formulate an experimental design optimization problem as before by introducing additional weights wy k 1 Thus we replace the model function h p og tr by weh p q tx with additional weight factors wy k 1
10. 300 301 302 Lindstr m P 1983 A general purpose algorithm for nonlinear least squares problems with nonlinear constraints Report UMINF 103 83 Institute of Information Processing University of Umea Umea Sweden Lioen W M de Swart J J B 1999 Test set for initial value solvers Release 2 1 CWI Amsterdam The Netherlands Liska R Wendroff B 1998 Composite schemes for conservation laws SIAM Journal on Numerical Analysis Vol 35 No 6 2250 2271 Liu X D Osher S 1997 Convex ENO high order multi dimensional schemes without field by field decomposition or staggered grids UCLA CAM Report 97 26 Dept of Mathematics University of California at Los Angeles Logan J D 1994 An Introduction to Nonlinear Partial Differential Equations John Wiley New York Logan J M 2001 Transport Modeling in Hydrochemical Systems Interdisciplinary Applied Mathematics Springer New York Lohmann T 1988 Parameteridentifizierung in Systemen nichtlinearer Differentialgleichun gen Dissertation Dept of Mathematics University of Bonn Lohmann T W 1997 Modellierung und Identifizierung der Reaktionskinetik der Kohlepy rolyse Fortschrittsberichte VDI Reihe 3 No 499 VDI Diisseldorf Lohmann T W Bock H G Schl der J P 1992 Numerical methods for parameter estima tion and optimal experiment design in chemical reaction systems Industrial and Engineering Chemistry Research Vol 31 54 57 Lorenz E
11. 4 0 1 0 2 9628 p2 2 0 0 0 1 3199 p 5 0 0 0 0 1332 p 1 0 0 0 1 6129 m 5 0 1 0 5 0223 v 1 0 1 0 0 1336 99 0 0 1 0 0 0009 Table 2 7 Optimal Start and Computed Solution for Cargo Problem v9 2 0 v9 23 0 and control functions u t u2 t so that the system follows the given trajectories as closely as possible at a given time T 2 Obviously we get a boundary value problem since we require x 0 a and x T b for i 1 2 3 Since we have to expand the differential equation 2 67 by additional state variables v1 V2 va to transform it into a first order system we get three further system equations d Vi 3 v2 3 vs Initial values are x 0 a and the unknown ones v 0 v fori 1 2 3 We try to compute these initial velocities so that the three nonlinear equality constraints z T b 0 are satisfied at an optimal solution for i 1 2 3 Exact parameter values starting values for DFNLP and the computed parameters are shown in Table 2 7 DFNLP stops after 38 iterations with a maximum constraint violation of 0 29 1078 Figures 2 19 and 2 20 display the computed trajectories and control functions respectively 2 5 7 Variable Initial Times The standard dynamical model described by ordinary differential equations or differential algebraic equations assumes that the initial time is zero even if measurement values are not available at t 0 At t 0 the solution is fixed in the form y p 0 cl
12. 474 475 476 477 478 479 480 481 482 483 484 485 486 quenching problems Numerical Methods for Partial Differential Equations Vol 16 No 1 107 132 Shiriaev D Griewank A Utke J 1997 A user guide to ADOL F Automatic differentiation of Fortran codes Preprint Institute of Scientific Computing Technical University Dresden Germany Shu C W 1998 Essentially non oscillatory and weighted essentially non oscillatory schemes for hyperbolic conservation laws in Advanced Numerical Approximation of Non linear Hyperbolic Equations A Quarteroni ed Lecture Notes in Mathematics Vol 1697 Springer Berlin 325 432 Shu C W Osher S 1989 Efficient implementation of essentially non oscillatory shock capturing schemes IL Journal on Computational Physics Vol 83 32 78 Silver S 1949 Microwave Antenna Theory and Design McGraw Hill New York Simeon B 1994 Numerische Integration mechanischer Mehrk rpersysteme Projizierende Deskriptorformen Algorithmen und Rechenprogramme Fortschrittberichte VDI Reihe 20 Nr 130 VDI D sseldorf Simeon B Rentrop P 1993 An extended descriptor form for the simulation of constrained mechanical systems in Advanced Multibody System Dynamics W Schiehlen ed Kluwer Academic Publishers Dordrecht Boston London 469 474 Simeon B Grupp F F hrer C Rentrop P 1994 A nonlinear truck model and its treat ment as a mu
13. 7 are inserted for the switching times When starting DFNLP from u 1 5 4 6 and T2 7 we obtain the results of Table 2 4 case 2 We are able to identify the friction coefficient and the implicitly given switching times It is necessary to add linear inequality constraints to satisfy zm Ta lt 10 22 Dry Friction u 1 5 2E 6 P 2E 6 10E 6 12E 6 t Figure 2 9 Function Plot for x t case Nit residual H T Ta T3 T 1 40 0 31 1078 0 009998899 2 57 0 57 10 0 009999998 5 78997 6 77911 3 33 0 62 107 Y 0 010000005 3 97642 3 97642 5 79262 6 77731 Table 2 4 Number of Iterations Residuals and Optimal Solution for Dry Friction Problem We cannot expect in general that the exact number of switching times is known If however too many switching times are defined in form of additional optimization variables there should be an overlay of final optimal values cancelling their influence To illustrate this situation we add two further switching times to our example in the same way as outlined before It is essential to add linear inequality constraints of the form 0 lt Ti lt To lt 73 lt T4 lt 10 to the least squares optimization problem to guarantee consistency of the model equations When starting DFNLP with the same termination tolerance used before and the initial values y 1 5 T 4 T2 5 T3 6 and T2 7 we obtain the numerical results shown in
14. Leugering G Schittkowski K Schmidt E J P G 2001 Modelling stabilization and control of flow in networks of open channels in Online Optimization of Large Scale Systems M Gr tschel S O Krumke J Rambau eds Springer Berlin 251 270 Gupta Y P 1995 Bracketing method for on line solution for low dimensional nonlinear algebraic equations Industrial Engineering and Chemical Research Vol 34 536 544 Guy R H Hadgraft J 1988 Physicochemical aspects of percutaneous penetration and its enhancement Pharmaceutical Research Vol 5 No 12 753 758 Haase G 1990 Dynamische Simulation einer Destillationskolonne und Entwurf einer Regelung Diploma Thesis Berufsakademie Mannheim 13 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 Hadgraft J 1979 The epidermal reservoir a theoretical approach International Journal of Pharmaceutics Vol 2 265 274 Hahn H 1921 Theorie der reellen Funktionen Springer Berlin Hairer E Lubich C Roche M 1989 The Numerical Solution of Differential Algebraic Systems by Runge Kutta Methods Lecture Notes in Mathematics Vol 1409 Springer Berlin Hairer E Norsett S P Wanner G 1993 Solving Ordinary Differential Equations I Nonstiff Problems Springer Series Computational Mathematics Vol 8 Springer Berlin Hairer E Stoffer D 1997 Rerversible long term integration with variable
15. OT 09 89S VG qq OET O8T 08 0 Ic LT OT y Y GT TT GG IG e o NA io N o Vad INQIHO OSO INdHO TANNVHO OTT NVHO IO NIVHO OHIO HO gang AVO ISAIVLVO GAH LYO HOLSVO OSSOSVO CHTvosvo TAACVOSVO dN OSVO ODUVO CULO UVO TALO YVO ANVO AOVAAVO YdAd TN dAd dAd T TILS ILLSH CTASSNUA IRC C IIHNOHS 9t Du 22 ponurjuoo SS GIG gS gS gS SS SS SS SS 030p oz 67 oz G6Z oz G6Z S6 pez vez vec iz 9ZT ESE 9cI SGT vez 617 617 GI 617 GI 617 Tez Cut 90 9cT fas MOP 3 u 3 u do Snonurjuo Suro30Jd JO got to uorjeurrojo uorje nuriog oArjeuioj e Topour uorje nurrs eoruroq 9pour uorjernuis eow sjureijsuoo K amp repunog pue 93e3s u3r4 uoryenbo s193 mg 1032901 Qojeq ruros Y ur uorjego1sos YPM uorjoeor x duroo ojeuoqd ns pououd exed umrpos pue op Aqop eurroj uoo43oq IWYS uorjoear x duroo s ro ds omy jo uorjmoeduro ursimpesuoururoo pue uore russe OATITJOdUMIOD uoneu tuo erguouodxo pue uorjdaosqe enbo Ota squourjreduioo 0A T Jopow io amp e r nur Aq squouoduioo omg Jo uorjd iospe oArjrjoduro soouooJo1d oje1jsqns oytsoddo tw eL1932 q OMT prey 9210 souof p reuuo enmu Ilo ur WOJE uooN Y pue UOSIY ue uooMjoq sorureu Ap uolsI O O013uoo pue Surxrub peq YL op eoseo UOIPPVIJXH cH THO ZOD OD Sutpngour suorjoeor 104314 o exed srsKpo1
16. Schitkowski 7 Path Plot Configuration Edit GNUPLOT Commands ri Overlays for Standard Plots o ql Grid Lines for Standard 3D Plots IT Initial X Angle for 3D Plots fi 0 Horizontal Increment of 3D Plots fi 0 Initial Z Angle for 3D Plots fi 5 Vertical Increment of 3D Plots fi 0 Figure 1 Configuration Form lv Name Default Contents system directory C EASYFIT EASY FIT esisn main directory with database help files etc editor EASY FIT Internal or alternative external editor for exam ple EDITOR EXE graphics system EASY FIT Internal or alternative external graphics system identified by string GNUPLOT Fortran compiler SALFORD Available Fortran compiler Insert ABSOFT WAT COM SALFORD LAHEY V FORTRAN MS_POWER INTEL or INTEL64 for Absoft Pro Fortran Wat com F77 386 Salford FTN77 Lahey F77L EM 32 Compaq Visual Fortran Microsoft For tran PowerStation Intel Visual Fortran com piler LA32 and Intel Visual Fortran 64 bit com piler EM64T respectively In case of installing an EAS Y FIT c Pesi n version coming with all object files proceed as follows First the submitted object codes of the numerical algorithms must be copied to a subdirectory with name OBJECT The object codes of the driving routines with names MODFIT FOR and PDEFIT FOR can be generated subsequently if necessary The easiest way is to edit dimensioning parameters in the Utilities menu and to let these modules be
17. Tx Bx DEXP B1 T DFIT 1 6 DEXP C1 T DSIN D T DFIT 1 7 T C DEXP C1 T DSIN D T DFIT 1 8 D C DEXP C1 T DCOS D T RETURN C C GRADIENTS OF CONSTRAINTS C 800 CONTINUE RETURN C C GRADIENTS OF RIGHT HAND SIDE OF ODE W R T Y C 900 CONTINUE RETURN END Example 9 2 SE The second example describes a single differential equation with three concentration values y k r y c y kay 9 2 Here the coefficients k ka and r are to be estimated The input file is the following one C EASYFIT PROBLEMS SE SE 4 Test Problem SE Demo Artificial Data 10 2 1994 Schittkowski 13 t NPAR 3 0 NRES 0 NEQU 0 NODE 1 NCONC 3 NTIME 14 0 NMEAS 1 NPLOT 50 NOUT 1 METHOD 1 OPTP1 100 OPTP2 8 OPTP3 2 OPTE1 1 0E 9 OPTE2 1 0 OPTE3 0 0 ODEP1 11 ODEP2 1 0 0 0 0 ODEP3 6 ODEP4 0 ODEE1 1 0E 9 ODEE2 1 0E 9 ODEE3 1 0E 6 Kp 1 0E 4 1 0E 4 10 0 Km 1 0E 6 1 0E 5 0 01 RO 100 0 150 0 400 0 SCALE 1 1 0 100 0 1 39 1 0 3 0 100 0 4 06 1 0 5 0 100 0 5 24 1 0 10 0 100 0 11 6 1 0 15 0 100 0 11 9 1 0 20 0 100 0 17 4 1 0 25 0 100 0 0 0 0 0 30 0 100 0 20 6 1 0 35 0 100 0 25 1 1 0 40 0 100 0 25 3 1 0 50 0 100 0 26 9 1 0 60 0 100 0 0 0 0 0 90 0 100 0 34 1 1 0 105 0 100 0 0 0 0 0 1 0 1000 0 16 5 1 0 3 0 1000 0 39 1 1 0 5 0 1000 0 57 4 1 0 10 0 1000 0 97 3 1 0 15 0 1000 0 118 5 1 0 20 0 1000 0 139 6 1 0 25 0 1000 0 159 8 1 0 14 30 0 1000 0 173 2 1 0 35 0 1000 0 180 3 1 0 40 0 1000 0 189 9
18. Vol 22 144 147 Olansky A S Deming S N 1976 Optimization and interpretation of absorbance response in the determination of formaldehyde with chromotropic acid Analytica Chimical Acta Vol 83 241 249 Osborne M R 1972 Some aspects of nonlinear least squares calculations in Numerical Methods for Nonlinear Optimization F Lootsma ed Academic Press New York Otey G R Dwyer H A 1979 Numerical study of the interaction of fast chemistry and diffusion AIAA Journal Vol 17 606 613 Otter M T rk S 1988 The DFVLR models 1 and 2 of the Manutec R3 robot DFVLR Mitteilungen 88 3 DFVLR Oberpfaffenhofen Germany Ou L T 1985 2 4 D degradation and 2 4 D degrading microorganisms in soils Soil Sci ences Vol 137 100 107 Pantelides C C Gritsis D Morison K R Sargent R W H 1988 The mathematical mod eling of transient systems using differential algebraic equations Computers and Chemical Engineering Vol 12 440 454 Papalambros P Y Wilde D J 1988 Principles of Optimal Design Cambridge University Press Park S Ramirez W F 1988 Optimal production of secreted protein in fed batch reactors AIChE Journal Vol 34 No 8 1550 1558 Peano G 1890 D monstration de Vint grabilit des quations differentielle ordinaires Mathematische Annalen Vol 37 182 228 Pennington S V Berzins M 1994 New NAG Library software for first order partial dif ferential equations ACM Transactions on
19. ub ps 2 4 0 1 2 101 for j Nae L Ne confer 2 92 Initial values are v p 0 vo p 2 102 97 that may depend again on the parameters to be estimated The system has ne components ie v vi Une Coupling of ordinary differential equations is allowed at arbitrary points within the integration interval and the corresponding area is denoted by the index 7 The spatial variable value zj some or all of them may coincide belongs to the th area l due 25 15 T or z z Thal respectively j 1 ne and is called coupling point Coupling points are rounded to their nearest line when discretizing the system The right hand side of the coupling equation may depend on the corresponding solution of the partial equation and its first and second derivative subject to the space variable at the coupling point under consideration To indicate that the fitting criteria h p t depend also on the solution of the differential equation at the corresponding fitting point where k denotes the index of a measurement set we use the notation a ma 1 gt holp t he p u p Tp t ue p zx t um p Ex t v p t t 2 103 and insert hy instead of hy into the data fitting function Again the fitting criteria may depend on solution values at a given spatial variable value witin an integration interval defined by the index ip The spatial variable z belongs to the i th integration area i e Le
20. uorjoeor eoruroqo t Topour 91 41 POXTIN uor jod equouruourauo pue uordunsuos uorgemdod yya pom suorjenbo moy domat Jo sts p derp YIM SIISE suorjenbo oam rqdoa3nou jo sis p derp YIM sramjsepq SJUYSIOM pue uorjoung yndut Y31M USIS9P eqguourriodxo 109989 101G Y9yeq paH uorounj qdur yya USIS op e3uourLiodxo sseurortq SULMOIS ouo YA 10J9B9IOIQ YOyeq pay punosbyo0q oo Ch Ch COO CO e coc co C So E ed Oo Om EN Oo n4 oO atii D G O C OS ot GO lt N cocco GC Ei r O GO CC H NY ri en GO hi bh GO ri 00 ri Po o0 LF CO CH lt M Jr JE Uu uL VI 09 gy 9G GE 0c er 0 c9 OF OT OT GG 08 08 O8T IT IT TT Ch Ch 0 NOD Y MN m m N GO O Gi Gi Gi P p hi ri OMS Gs c u NADOYHLIN OULIN ATULIN SHY ON ULSA TAIN ATOSNIN AV ILLIQIN HOLVIAN NOILON dSL LON NOOIN AYALSION TILO NIN LANDININ CVLAN WIN TV LAW WIN dOdXIIN ELVY XIN LV XIN LIV XIN C RIOAANTIA CHA TIN TITIN ZNOWDOIIN AMOYIOIN IWDU 32 ponurjuoo ouou leog a a a a a a a eS 9ZT IS soe Tope eS v67 euou 6ps Let euou ore ler x yay cs cs IS LS a A cs vez GS cs 0S a A eS GS er kee 030p fas ut9 pue JUBVOT juopuedop oeurn YM 10350991 EULIOQ3OSI UON U00 YIM ureurop Aouonbo g ur Me VLIOJVUL 21 S9 900STA Ie9UI UON uio qoxd o1juoo eurndo s 09 oseqgo
21. 2 2 2 7 13 Parameters to be Estimated 3 7 1 4 Input Type of Model Functions esla NEE 3 7 1 5 Numerical AnalysiS lt sn 4 TAS Optimization Tolerances idos bee esa q UR Sy Ew N 6 TO GBOBDBE S euch eoe des a eo kak dokgboR a a 11 TLS Numberol Plot Points A 1 REGUM eh w q e ee XU EE e N 11 CLI Saline Plop PD 11 LLO Expernmental DAN e du xa doy RE b Rum meos ew xh oe S S QG U 11 Ta 8 1 8 2 8 3 8 4 8 5 8 6 Br TAAL Data Fitting NODE Lun wk Wk ek doane Se Boe RO RR ee 12 Model Dependent Information 13 7 21 Model Data for Explicit Functions lt lt lt lt lt 55 13 7 2 2 Model Data for Steady State Equations 16 7 2 3 Model Data for Laplace Transformations 20 7 2 4 Model Data for Ordinary Differential Equations 21 7 2 5 Model Data for Differential Algebraic Equations 26 7 2 6 Model Data for Time Dependent Partial Differential Equations 33 7 2 7 Model Data for PDA S 4 4444 c don beh Sr oh REO 40 8 Menu Commands 1 File Command ccros eig e bar 1 Fan Command e ce ee BERGER III 5 Start Command e ss aca 3 2 S e aa e a A 6 Report Command se dca w eh GG m EES eRe Dawe aD ede e A g a 7 Data Command A ires ee aoa E k pun Ge musu Q sU Ae A AC 10 Delete Command e zoom mo BUE wna ub Rm sP g SE EGE Ee eG Ox 14 Make Command o e ess s s be REAR S QO b RS Q ERA Q S
22. 24 0 8 162 1 0 NLPIP 0 NLPMI 0 NLPAC 0 0 11 OO OO OO Q Q CH NBPC 0 The code DFNLP computed a solution in 24 iterations and the optimal fit is shown in Figure 9 1 The corresponding Fortran subroutine SYSFUN can be implemented as follows Q Q Q Q SUBROUTINE SYSFUN NP MAXP NO MAXO NF MAXF NR MAXR X Y T CONC YP YO FIT G DYP DYO DFIT DG IFLAG IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X MAXP Y CMAXO YOCMAXO YPCMAXO G MAXR FIT MAXF DYO MAXO MAXP DYP MAXO MAXP MAXO DG MAXR MAXP DFIT MAXF MAXP MAXO BRANCH W R T IFLAG IF IFLAG EQ 0 RETURN A BT A1 B Bi C ci D GOTO X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 100 200 300 400 500 600 700 800 900 IFLAG RIGHT HAND SIDE OF ODE 100 CONTINUE RETURN INITIAL VALUES FOR ODE 200 CONTINUE RETURN FITTING CRITERIA 300 CONTINUE FIT 1 A DEXP BT T 0 1 DEXP A1 T B DEXP B1 T C DEXP C1 T DSIN D T RETURN CONSTRAINTS 400 CONTINUE RETURN 12 Q GRADIENTS OF RIGHT HAND SIDE OF ODE W R T X AND Y C 500 CONTINUE RETURN C C GRADIENTS OF INITIAL VALUES OF ODE W R T X C 600 CONTINUE RETURN C C GRADIENTS OF FITTING CRITERIA W R T X C 700 CONTINUE DFIT 1 1 DEXP BT T 0 1 DEXP A1 T DFIT 1 2 T A DEXP BT T DEXP A1 T DFIT 1 3 Tx Ax DEXP BT T 0 1 DEXP A1 T DFIT 1 4 DEXP B1 T DFIT 1 5
23. 3 227 241 Knabner P van Duijn C J Hengst S 1995 Crystal dissolution fronts in flows through porous media Report Institute of Applied Mathematics University of Erlangen Ko D Y C Stevens W F 1971 Study of singular solutions in dynamic optimization AIChE Journal Vol 17 160 166 Kojouharov Chen B M 1999 Nonstandard methods for the convective dispersive transport equation with nonlinear reactions Numerical Methods for Partial Differential Equations Vol 16 No 1 107 132 Kopp R Philipp F D 1992 Physical parameters and boundary conditions for the numerical simulation of hot forming processes Steel Research Vol 63 392 398 Tf 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 Kowalik J 1967 A note on nonlinear regression analysis Australian Computational Jour nal Vol 1 51 53 Kripfganz J Perlt H 1994 Arbeiten mit Mathematica Carl Hanser Oldenburg Kuhn U Schmidt G 1987 Fresh look into the design and computation of optimal output feedback controls for linear multivariable systems International Journal on Control Vol 46 No 1 75 95 K hn E Hombach V 1983 Computer aided analysis of corrugated horns with axial or ring loaded radial slots Report Research Institute of the Deutsche Bundespost Germany Kung AHC Baugham R A Larrick J W 1993 Therapeutic Proteins W H Fremmen New York Kuzmi
24. 30949 1911105 uo poseq ueiqur ur YINOIYY uorsngt q seo1e OM y3n0 uorsngtp ojezjsqng suiojs s eoruroqo uorsngrp uorjoeax UL Surjsinq juo33rurro ur poonput SISO G000 0 sda uorjngos 3oexo qt uorjenbo s 193g SnoostA Grp ergrur uueurory yya uorjenbo s 1931mg yur PIOSTAUT oy ur uorjenbo s 193 mg enur uorjnjos 3 exo yya uorjenbo s 1oS4ng snooSsrtA TO OO uorjnjos 3oexo HL uorjenbo s tle31ng SNOIN UOTJNTOS 3joexo YIM uorjenbo s 103mg or oqedeq uumnoo o qqnq ur 193 A Jo oyeydn uoS xo orureuX q IoJoRaI O1q uurmn oo qqng AYLINDNOS oArjeALIop JO d SUTUIDAOS uoryenbo so oqog e q uorsngrtp YIM 1oje ossnagg IOje ossnuq euorsuoutrp oA T erpour snood ut oda out 9NSSIY ureiq ur euourououd ode dn mo q gn uorjenbo oroqered pojedouoso T Iotnreq uredq poo q Uorjoeol WIYOI o3e1jsqns ANO suorjoefur OM YIM s qrs Surpurq pue repnoseAer xo eurse q Apnys pes1odsrp joosur soyoyed paej oo jo SABIE TROUT pozean ur s 399q LA slwoq G Jo suris Y3NOIYI ire Jo uorsngr q uorsngjitp IRCH YM 1039891 re nqnj snonurquo ured oprure1e ue ogur uorjerxjouod 1978M UOTPOVOI eoruroqo YIM UON Aoje umrurumge ur gurjonpuoo 3e9u OMU puno4bDsovq o oooo coo CC CH OC cn Ch Ch Ch CC Ch CH o oo oo CH Y A Oo NOOO OO OOO OOO Ch Ch Ch Ch Ch CH O CC Ch CH lt t NA MOA ei r c c ri CN N lt O GN ed rd N eS Sst es ed ort 8 GI S6T SOT 0G vr GE 0c UI VG 80I OT OT 0cI 03 ST 0
25. 440 NO CHAO NO TAJO NO ODITO TIO SASOMHO PXH DO EXT OO TLO LSdO CAHdASHO TAYASAO LLRLLON CVLN TV LN ULSON NDT NON OST NON AHAIA NON GIA NTIN TIA NIIN AGO IN ULSO IN 9 Du 33 ponurjuoo a q q ug Trop cs 66 os IS os el Joes CI qS OST q x IT a LS Gs got Gs vec IS 971 Gs 971 IS Gs VEZ Gs EST a ag q rr eer ouou JET Gs Lt CI euou 908 68 es 09 logs 030p fas SUIT erarut sonpeA erjtut o qeLreA YILM Topour R Som 21 erjrur enprarpur o qeIIeA Ou opour orureu Ap ooeuLreuq sorureu p uorje dod perqoaorur yor dxa yya uorgepe1dsop oprorgsoq uip roruod Jo SISOYJU SOIQ IO 103uourro qo3eq po4 uorge nurio 0Q xopur umnpuod ure q unnpued onserq sprqde uve od jo azis uorye ndoq syeod dregs va ACO BUS Od Jo uonezruoSo1pKu 91 48169 Jo SISATeUR IUA JAg 19p10 pug uorjoeor pue uorsngrp o pry req soye1 dnpeoiq pue oouooso eoo Sgurpn our oouepeq uorjepidoq o1oudsourje UL sor upi Uozo xXuej pojeioe UL ur UOIj2991 UOI epIX Uorjeprxo orjyeur amp zuo JO 1032991 YUL Y JO OLI Y sorureu amp p uorjeioe pue oxejdn uog x WOTJOVOL org Aqpe3eo snoouosod03j9Qu ur suorjeros IOTABUYOY 1099891 JUL Surje I2s Sojerpourioqur oA JO uorjerjuoouoo ou JO WOTYLTTIOSC soouengur juorsue1 TILA urojs As Surje ISO UOTJP IOSO eorumou 1038U039 0 uorjoeor ro surjoqeqz Aosnpogq o13uoo peurrjdo Joie 11 1020 SUIT VINVITUTA U
26. 9 LG VG 0 IS 9 EST OT 0 IG 96 GL TOT 06 OV GG io oud oD Y hi CH DML P si aO a0 cM CN CV CN CV CN c bh D CN Co NN SSKT AVL vVSIC AVL SId JVL cSId AVL ISIC VL XOUddNs TXOUdd NS AOT IOS SITLLEANS NOV ans CAHILS TAMILLS OW AALLS Ad AALLS AILS SLVLS SUVLS UVLs Ca DVLS TA vis ULSOIVLS XHALVHHSS dani ss ONTUdS DHGOWgds LOOS XH TIVINS IWDU 38 ponurjuoo IS gS gS gS gS gS gS SS SS gS gS gS gS IS SS SS IS 030p soe car ca cay vez 971 931 VEG 9cT VEG vez vez Tre vec 9cT EST EGG YEG VET VET VET 866 86 866 foa snip V Jo uon 919X9 pue UOISIOAUOD or oqe3our ayxeydn 103 opour quourjreduio 013u00 9A9 JULI oA T WLIS euorgrppe YIM 3e3souroqo 3 e1s oA T suistue S10 uooAjoq uonmn duroo eriqrmbo Surguequooqur YIM uorjor xo qo3eq ojnpos oA43 xo duro osuodsori 1e3soptq mn Dot Surxrur ue3 oqn3 eopr uoN 1039891 re nqn1 e ur o go1d uorjerjuoouoo erxy suornoe 21 Iopio q3u YPM Soles ur SULI JBT 1032 o1 TO ormeudq 1039891 89TUI9YI Y Jo s1o3ourered DIJQUTA JO UOLYBUILISH umouejno UOIOASOYILA Jo 414018 P9INJINIJSU 89913 JO YIMOLL uourosodrT Moya c 09e7 ur 1rAeurpugp umnse qurtA UTA pue uourosodrT JNOYIIM Z OoVD ur 1rAeurpugp pt POLIS snonurquoo o8 3s 0A Jo OIJUO9 Teurridoqns owg uorjeor orKpeqeo Aq uorjyeuoSo1p q ouon oT ULSO Jo Ayrq
27. 986 LET UOTILA eorur uO IOYeTIOSO eoru JAg 1op1o PIE pouueuo Y ut MOLT puueyo Suo e ogur uorjoofur Surmp prng Y JO MOTT UOIj2 9I ureuo 9 QISI9A91 IDPIO 9SM J oulrdo1 STORY Y UI HOI 2LI329 d e qqnq Suryeyaeo 1030991 re nqnj e ur ouozuoq o3 euequodo oAo 4ngour Jo puo q 3sAq egeo p euorjoungtq opripKque 919998 Jo SIS TOIPAY or Apege IO 103589 poje A3ooe Jo uorjrsodurooop qojeq suorjoeor Ferguonbos YPM S 10J9891 201 JO opeoser uoryenbo umSurp sn A 1eo ory a3e109s ouo YALA sadid ut MOTT uorjenbo 13e991y sedid ut Mog Jo opeoseo 931038 Ap K amp ioje1oqe ur Ajrprumu dry YONI 09 drys WO saourequoo SULLIOJSURIT o1n reg 3 eo1q o qissod Ypa re JO uorjelo o2o V Ted Y JO UOI3 I9 000 V 1032991 TROTMY V Jo s1o3ourered orjoun JO VOLYBUITISH soweo uro3s s3001 oSeqqeo Ou Jo 34019 Add Jo uonrsoduroo op opoul eurrou uro qo1d onfea Arepunog 1 pio yyy x dtuoo uro qoad NJLA Arepunoq TROUTTUO NT wo qoid n eA Arepunog srtooU u A uumgoo uorje rsrp qoyeq Areurg Ss100g ouru uumpqoo uorye rstrp qo3eq reug 10ye ossnIg UOV re noo our rn TA 10ye ossnIg UOV re noo our rn TA ojejrs oArjoeorpez Jo uorjoofur snog puno4bDsovq CO ONANA MO TF OOO Ch CH CO CO CO Gi Gi CO CH O CC Ch Ch ei CH oi GOON N OO C S ONO Eet 1 eat EN C O CX dO S e C 00 N NA NN O 710 c CN N CX E NON f MI O CN 19 CO mi Ee BN 66 0G Gs OV OST VG 88 0
28. C 200 CONTINUE YO 1 B Y0 2 B A 1 B B RETURN e C FITTING CRITERIA C 300 CONTINUE FIT 1 Y 1 20 Q Q FIT 2 Y 2 RETURN CONSTRAINTS 400 CONTINUE RETURN GRADIENTS OF 500 CONTINUE RETURN GRADIENTS OF 600 CONTINUE RETURN GRADIENTS OF 700 CONTINUE RETURN GRADIENTS OF 800 CONTINUE RETURN GRADIENTS OF 900 CONTINUE DYP 1 1 DYP 1 2 DYP 2 1 DYP 2 2 RETURN END RIGHT HAND SIDE OF ODE W R T X AND Y INITIAL VALUES OF ODE W R T X FITTING CRITERIA W R T X AND Y CONSTRAINTS RIGHT HAND SIDE OF ODE W R T Y 0 0 1 0 2 0 A Y 1 Y 2 1 0 A 1 0 Y 1 2 21 1 9 1 8 1 7 1 6 1 5 1 4 1 3 1 2 1 1 Figure 9 3 Final Trajectories for Problem VDPOL 22 e data Y 0 1 02 03 04 05 06 07 08 0 9 t y datal I Z S Reel 0 1 02 03 04 05 06 07 08 09 Example 9 4 RECLIG10 To illustrate the implementation of a steady state system we consider the following example that is similar to a receptor ligand binding study with one receptor and two ligands 21 1 p122 pez3 p3 0 zo 1 un pia p4 0 9 4 23 1 poz t 0 The system parameters are 21 22 and za and the parameters to be e
29. Dennis J E jr Gay D M Welsch R E 1981 Algorithm 573 NL2SOL An adaptive non linear least squares algorithm ACM Transactions on Mathematical Software Vol 7 No 3 348 368 Dennis J E jr Gay D M Welsch R E 1981 Algorithm 573 NL2SOL An adaptive non linear least squares algorithm ACM Transactions on Mathematical Software Vol 7 No 3 369 383 Dennis J E jr Heinkenschloss M Vicente L N 1998 Trust region interior point SQP al gorithm for a class of nonlinear programming problems SIAM Journal on Control Vol 36 No 5 1750 1794 Deuflhard P 1979 A stepsize control for continuation methods with special applications to multiple shooting techniques Numerische Mathematik Vol 33 115 146 Deuflhard P Apostolescu V 1977 An underrelaxed Gaufi Newton method for equality con strained nonlinear least squares Proceedings of the IFIP Conference on Optimization Tech niques Part 2 A V Balakrishnan Thoma M eds Lecture Notes in Control and Information Sciences Vol 7 22 32 Springer Berlin Diener I 1986 Trajectory nets connecting all critical points of a smooth function Mathe matical Programming Vol 36 340 352 Dietrich E E Eigenberger G 1996 Compact finite difference methods for the solution of chemical engineering problems in Scientific Computing in Chemical Engineering Keil Mackens Voss Werther eds Springer Berlin Dobmann M Liepelt M Schittkowski K 1995 Algorithm 746 P
30. In case of parameter estimation in ordinary differential equations it is possible to select either an implicit solver for stiff equations RADAUS an explicit solver for non stiff equa tions DOPRI5 or an explicit solver with internal numerical differentiation for non stiff equations see Benecke 36 All codes apply Runge Kutta method of order 4 to 5 the last one with additional sensitivity analysis for the evaluation of derivatives For more details see Hairer Norsett and Wanner 197 or Hairer and Wanner 199 respectively Arbitrary linear or nonlinear constraints can be taken into account For implicit methods gradients of the right hand side of the differential equation can be evaluated analytically using either user provided derivatives or automatic differentiation The implicit code uses dense output i e the integration is performed over the whole interval given by first and last time value and intermediate solution values are interpolated In this case gradients with respect to the parameters to be estimated are obtained by external numerical differentiation The explicit algorithm is capable to evaluate derivatives of the solution of the ODE internally with respect to the parameters to be estimated i e by analytical differentiation of the Runge Kutta scheme It is possible that the right hand side of an ODE is non continuous subject to integration time for example if non continuous input functions exist Especially in case
31. L h p s c of an unknown function bin t c we compute the coefficients min i q kt 2 pue epe q EI DI EJ QR 9 4 1 k i 1 2 which are independent of H and which can be evaluated before starting the main procedure where function values are to be computed Then iln2 In2 Y h p t c 2 5 H p vg 4 2 i 1 is a numerical approximation formula for h The parameter for controlling the accuracy is the number q When working with double precision arithmetic it is recommended to use q 5or q 6 Any smaller value decreases the required accuracy any larger value introduces additional round off errors For the practical models we have in mind the numerical instabilities induced by oscillating function values do not appear A particular advantage of the formula is that the derivative of h with respect to the parameters to be estimated are easily obtained from the derivatives of H 4 4 Ordinary Differential Equations The parameter estimation program MODFIT that is executed by EASY FIT V tc Pesisn ag an external executable file through the shell feature of the Microsoft Visual Basic language organizes data and evaluates the fitting functions with additional features for example to process input data in a special format to provide problem dependent output or to gen erate plot information The underlying dynamical model consists of a system of ordinary differential equations with initial values
32. MAXP NO MAXO NF MAXF NR MAXR X MAXP Y MAXO T C YP MAXO YO MAXO FIT MAXF G MAXR DYP MAXO M1 DY0 MAXO NP DFIT MAXF M1 Number of parameters in array X Dimensioning parameter must be greater or equal to NP Number of equations of dynamical system Dimensioning parameter must be greater or equal to NO Number of fitting functions in the parameter estimation problem that corresponds to the number of measurement sets Dimensioning parameter must be greater or equal to NF Number of constraints of the parameter estimation problem Dimensioning parameter must be greater or equal to NR When calling SYSFUN X contains the NP coefficients of the ac tual variables to be estimated X is not allowed to be altered within the subroutine When calling SYSFUN Y contains the NO coefficients of the dif ferential equation on the right hand side Y is not allowed to be altered within the subroutine Time variable Concentration variable Function values to be evaluated in case of IFLAG T for right hand side of a dynamical system Function values to be evaluated in case of IFLAG 2 for initial values of the dynamical system Function values to be evaluated in case of IFLAG 3 for the NFIT fitting conditions in a parameter estimation problem Function values to be evaluated in case of IFLAG 4 for con straints in the parameter estimation problem Gradient values to be evaluated in case of IFLAG 5
33. Simultaneous optimization nad solution methods for batch reactor control profiles Computational Chemical Engineering Vol 13 49 62 Dahlquist G Edsberg L Sk llermo G S derlind G 1982 Are the numerical methods and software satisfactory for chemical kinetics in Numerical Integration of Differential Equations and Large Linear Systems J Hinze ed Springer Berlin Daniel C Wood F S 1980 Fitting Equations to Data John Wiley New York Davidian M Giltinan D M 1995 Nonlinear Models for Repeated Measurement Data Chap man and Hall London Davis H T 1962 Introduction to Nonlinear Differential and Integral Equations Dover de Saint Venant B 1871 Th orie du movement non permanent des eaux avec application aux crues des rivi res et Vintroduction des mar es dans leur lit Comptes Rendus Academie des Sciences Vol 73 148 154 Denbigh K G 1958 Optimum temperature sequence in reactors Chemical Engineering Sciences Vol 8 125 132 106 107 108 109 110 111 112 113 114 115 116 117 118 Dennis J E jr 1973 Some computational technique for the nonlinear least squares problem in Numerical Solution of Systems of Nonlinear Algebraic Equations G D Byrne C A Hall eds Academic Press New York London Dennis J E jr 1977 Nonlinear least squares in The State of the Art in Numerical Analysis D Jacobs ed Academic Press New York London
34. We do not require the evaluation of all boundary functions Instead a user may omit some of them depending upon the structure of the PDE model for example whether second partial derivatives exist in the right hand side or not In addition the partial differential equation may depend on the solution of a system of ordinary differential equations v IR given in the form j G p u p j t Un lp Xit Usp Sat 9 t 45 for j L ne where u p x t is the solution vector of the partial differential equation Here x are any x coordinate values where the corresponding ordinary differential equation is coupled to the partial one Some of these values may coincide When discretizing the system by the method of lines they are rounded to the nearest neighboring grid point The corresponding initial values v p 0 vo p may depend on the parameters to be estimated Each set of E data is assigned a spatial variable value x zr R k 1 r where r denotes the total number of measurement sets Some or all of the z values may coincide if different measurement sets are available at the same local position Since partial differential equations are discretized by the method of lines the fitting points x are rounded to the nearest spatial grid point or line respectively The resulting parameter estimation problems is min har Wiki wh Alp u p Ex te us p Ex ti Une Py Zk ti U p ti ti wp pedi gp 2
35. aag XIa OVNIA IWDU 92 gS gS gS gS gS gS gS gS uou gS IS gS GS GS SS SS SS SS SS 030p p nuruoo cer cer eer mm O e cP ar N 061 061 061 Oct LEG Oct Oct OTS SPS EGT OSV fas ulojs S 1899 ot oq iod AH urojs S 189 ot oq iod AH UIL9IJS S 1899 ot oq iod Arq ulojs S 189 ot oq iod Arq sydd ot1oquodAg 19p 10 3s1g suoryenba uorjooApe om jo toys kg SOAVM QUISO SULYLUIOYe 1opio puooos Jo uorjenbo otroqadodARq suorjrpuoo amp repunoq orporied n uoryenbo otr oq iod Arq Ire pue Tenogeu otdooso18Aq jo 1o4ep Arepunog y3n0 1 1998M Jo uorsngt q Zo Tod ordoosoi1S q ozur 1936A JO uorsngr q UOTJP NULIOJ xnp surojs s OIpAy Jo sorureu p pmp 10 uoryenbo jueuoA 98 189S SULOIS S OIP Y Jo sorureu Ap ping 10 uoryenbo queuoA 4S Stop eLIos omy suro3s s OIDAT jo sorureu p pmp 10 uorjenba jueuoA 98 suro3s s OrpAy jo sorureu amp p pmp 10 uoryenba queuoA 4S umrp ur snorod Jo uumngoo uomnsuogrptumH 109139 uorjsnquioo woy ut qoid ode 30H ULIOU umurxeu pue e3ep y9exo YIM uoryenbo matt lojoure rd uorsngrp 3uopuodop ourn YIM uorsngtp 29H uomnn os yoRXo pue uorjenbo Ayrargrsuos ouo YL uorjenbo eog suorjenbo AYLAT ISUOS M ILM UOTJONpUOD JLIH si oyeurered yuepunpel YIM uorjenbo map si oyeurered yuepunpel YIM uorjenbo map d uuewzyjog uejpyg jo uonrpuoo Arepunoq reouruou ujr UoTenbs je9H 19p10 P
36. and Y and store results in DFIT Evaluate gradients of constraints with respect to X and store results in DG Evaluate gradients of the right hand side with respect to Y only and store results in DYP if an implicit ODE solver is executed in case of IFLAG 9 There are some files with names MODFUN_E FOR and MODFUN O FOR distributed together with EASY FIT c Pesis that contain source codes for very simple examples to illustrate the usage of user provided Fortran code The input of data and model functions is to be outlined in addition by some examples Example 9 1 DFE1 The first one consists of an explicit model function given in the form h p t a exp t 0 1 exp a4t bexp bit cexp cit sin dt The input file must contain then the following information C EASYFIT PROBLEMS DFE1 DFE1 1 Test Problem DFE1 Demo Artificial Data 10 10 2 1994 Schittkowski x t NPAR 8 0 NRES 0 NEQU 0 NODE 0 NCONC 0 NTIME 10 0 NMEAS 1 NPLOT 50 NOUT 1 METHOD 1 OPTP1 100 OPTP2 8 OPTP3 2 OPTE1 1 0E 10 OPTE2 1 0 OPTE3 0 0 ODEP 1 0 0 0 0 0 ODEP2 0 ODEP3 0 ODEP4 0 ODEE1 0 0 ODEE2 0 0 ODEE3 0 0 a 0 0 1 0 1000 bt 1 0E 5 0 1 1000 al 0 0001 0 001 1000 b 0 001 5 0 10000 bi 1 0E 5 100 0 10000 G 0 0 0 0 1000 cl 1 0E 5 0 1 1000 d 0 0 0 00001 10000 SCALE 1 0 0 25 00 1 0 3 0 15 51 1 0 6 0 16 08 1 0 9 0 16 11 1 0 12 0 15 48 1 0 15 0 14 24 1 0 18 0 12 50 1 0 21 0 10 41 1 0 22 5 9 299 1 0
37. and are given for each integration area separately Next n coupled differential equations followed by the coupled algebraic equations are specified in the order given by the series of coupling points i e the functions G p U Us Ure V t J 1 ne where the state variable u is evaluated at a given discretization line together with its first and second spatial derivatives The corresponding initial values of the coupled ordinary differential algebraic equations at time 0 must be defined vip j 1 n in the same order Then n Dirichlet transition and boundary conditions must be set in the order given by the area data first left then right boundary ci p u v t cs p u v t where function values of u at the left or right end point of an integration area are inserted Subsequently transition and boundary conditions for spatial derivatives must be de fined in the order given by the area data i e the functions 6j p u u v t Cay p U Uz V t Again the function values of u or u at a suitable end point of an integration area are inserted Moreover r fitting criteria have to be given any functions hi p U Uz Use v t h p U us Ure v t u is defined at the corresponding spatial fitting point The final m functions are the constraints gi p 9m p if they exist They may depend on the parameter vector p to be estimated Any other functions are
38. and the more accurate is the curve Note however that in case of an ODE DAE and PDE only one integration is performed over the time axis and the required solution values are retrieved from intermediate iterates of the algorithm by so called dense output i e by internal interpolation Thus the calculation time is not increased dramatically in case of a larger value The number of plot points must not be bigger than the parameter MAXPLT in the include file of the corresponding analysis code and must not exceed the maximum number of time values 7 1 9 Logarithmic Plot It is possible to require a logarithmic scale of the time axis in function and data plots In this case a plot is generated from the first to the last measurement value not beginning at t 0 Consequently the corresponding time values must be positive 7 1 10 Experimental Data If denotes the number of experiments then one has to insert records containing the following data Column headed by time Input of measurement time t i 1 l A set of experimental time values has to be repeated for each concentration value If more than one measurement set is available we assume that experimental data are available for all time values Otherwise experiments can be eliminated by zero weights If the smallest time value is less than zero n then the integration of an ordinary or partial differential equation is started at the point given Otherwise integratio
39. ap 20 qasin 7z 2 86 v p z 0 0 for all x 0 1 Dirichlet boundary values are the same as before see 2 80 and the parameter vector p D a is to be estimated subject to the same simulated experimental data computed for Example 2 17 When starting the same least squares and integration algorithms as before we get an identical solution after 5 iterations The initial values are not consistent but are easily computed within machine accuracy in two iterations by the nonlinear programming code NLPQLP of Schittkowski 427 440 449 Stopping tolerance is set to 107 The maximum error of the algebraic equation along the lines x 0 25 z4 0 5 and zs 0 75 is 0 11 1078 The corresponding plot for the algebraic variable v p x t is shown in Figure 2 25 If on the other hand the initial values for v are changed to u p r U asin zz v p z 0 acos 7z we get theoretically consistent initial values However spatial derivatives are approximated numerically so that we have to relax the termination tolerance for executing NLPQLP to 107 according to the discretization accuracy to avoid re calculation of consistent initial values of the discretized DAE 47 Heat Transfer TN AAR SAI ga mW Wu M W NW Wu i Wu Wad NR ANN Qn WII AN NAM Wou If N N Wy WW WW W Figure 2 25 Surface Plot of Algebraic State Variable 2 6 3 Flux F
40. for the right hand side of the dynamical system for variables X and Y and for Y only in case of IFLAG 9 where M1 MAXP NO Gradient values to be evaluated in case of IFLAG 6 for initial values with respect to variable X Gradient values to be evaluated in case of IFLAG 7 for the fit ting criteria of the parameter estimation problem with respect to X and Y where MI MAXP NO DG MAXR NP Gradient values to be evaluated in case of IFLAG S for the con straints of the parameter estimation problem with respect to X IFLAG Flag defining the desired type of calculation IFLAG 0 IFLAG 1 IFLAG 2 IFLAG 3 IFLAG 4 IFLAG 5 IFLAG 6 IFLAG 7 IFLAG 8 IFLAG 9 Subroutine SYSFUN is called with IFLAG 5 only if an ODE solver with internal nu merical differentiation is used i e IND DIR On the other hand derivatives are needed only Execution of SYSFUN before requiring function or gradient values e g for preparing common s Evaluate right hand side of dynamical system and store results in YP Evaluate initial values of dynamical system and store results in YO Evaluate fitting criteria and store results in FIT Evaluate constraints and store results in G Evaluate gradients of the right hand side with respect to X and Y and store results in DYP Evaluate gradients of initial values with respect to X and store results in DYO Evaluate gradients of fitting criteria with respect to X
41. koa OS 15 Utilities Command LC 16 8 8 External Usage of Numerical Codes 91 MODFIT Explicit Model Functions Dynamical Systems ODE s and DAE s 1 92 PDEFTT Partial Differential Equations 2 6 om GR RR 28 10 Test Examples 1 10 1 erster Model Functions u c s s gt 99 eb W XY RES RE EO QU S 3 10 2 Laplace TESIS gt A eom ea ce W be et we oe hee ee he EES 14 10 3 Steady State Equations e ease Ses BORE we SS Owe oS e W 15 10 4 Ordinary Differential Equations lt lt 4 2 542444 44484 844 08 18 10 5 Differential Algebraic Equations lt 24444 5 044540 64 24404 41 10 6 Partial Diferential Equations lt ce s s A s w s ala RO ea t Re POR RO di 45 10 7 Partial Differential Algebraic Equations lll 60 0 1 Installation EASY FITdcPesis consists of a database containing models data and results and of underlying numerical algorithms for solving the parameter estimation problem depending on the mathematical structure i e MODFIT parameter estimation in explicit functions steady state equations Laplace transforms ordinary differential and differential algebraic equations PDEFIT parameter estimation in one dimensional time dependent partial differential equations and partial differential algebraic equations By the following notes the system installation and hardware requirements are outlined 0 1 1 Hardware and Software Requirements Installation of EASY FIT 4 Pess requires 95 MB on hard disk plus
42. minimizer of 3 2 and y gt 0 be given 1 Compute the lowest eigenvalue Amin of Ik and a corresponding eigenvector Umin IR Qa Em Uu PL LUUD 1 2 If Amin gt 73 then stop The required significance level is reached Y 3 Determine jg with al max ul eliminate the jo th row and column from Du denote the resulting matrix by rere and let Jai Jk U jo 4 Ifk 2n 1 then stop a further reduction is not possible 5 Replace k 1 by k and repeat from Step 1 After termination the indices in J represent the significance levels of the parameters Level 1 corresponds to the first eliminated variables level 2 to the second etc The final level can be assigned to several parameters indicating a group of identifiable parameters Possible conclusions are to add more experimental data or to fix some parameters for subsequent evaluations Thus the determination significance levels are part of the experimental design process to validate a parameter estimation model Example 3 2 LKIN _A3 _A4 A linear ordinary differential equation describes a kinetic process in the form Y hWw 0 D Yo k y kaya Yya 0 0 A 95 confidence region as outlined in the previous section is shown in Table 3 2 i e 6 20 s di see 3 6 The estimated error variance is 0 41 107 the maximum correlation is 0 57 and the covariance values are sufficiently small see also Figure 3 2 Now we introduce some addit
43. nchen Institut f r Angewandte Mathe matik und Statistik van den Bosch B A J 1978 Identification of parameters in distributed chemical reactors in Distributed Parameter Systems W H Ray D G Lainiotis eds Marcel Dekker New York Basel 47 134 van den Bosch P P J van der Klauw A C 1994 Modeling Identification and Simulation of Dynamical Systems CRC Press Boca Raton Ann Arbor London Tokyo van Doesburg H De Jong W A 1974 Dynamic behavior of an adiabatic fixed bed methana tor in Advances in Chemistry Vol 133 International Symposium on Reaction Engineering Evanston 489 503 34 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 van Duijn C J 1989 Flow through porous media DFG SPP Report No 135 Dept of Mathematics University of Augsburg van Genuchten M T 1980 A closed form equation for predicting the hydraulic conductivity of unsaturated soils Soil Science Society of America Journal Vol 44 892 898 van Genuchten M T Wierenga P J 1976 Mass transfer studies in sorbing porous media 1 Analytical solutions Soil Sciences Society of America Journal Vol 44 892 898 van Kan J J I M Segal A 1995 Numerik partieller Differentialgleichungen f r Ingenieure Teubner Vande Wouwer A Saucec Ph Schiesser W E 2001 Adaptive Methods of Lines Chapman and Hall CRC Boca Raton Varah J M 1982 A spline least squares
44. opour I9 RTIADIN uorjoeor xo duroo YAM IORI eUurrou30sT uoryepnurs Are rrdeo 1041 s19Z eg jo Axe ide 1032891 YUL pois snonurquoo JO OIJUO 1039891 YUL POLIIYS SNONUTJUO9 Jo Toto jurod3og o durexo x reurqouoq Y LSO opeoseo NULI P9ITLIS SNHONULJUO SUOTJOVOI ISIDADA MOYPM 3uouroe doi HN uonrp duroo 109 Sur ooo YIM YUL peris snonurquoo IOPIO ISM J 1039891 polis A snonurtquo ULID Iourejuoo Y Jo o13uoo peun do UO01J9891 PEWO 1oddoo uo poqiospe onjsrour 10 opour 1o amp e rg n A uorjdaosop yya 1oddoo uo poq iospe o1nj3srour 10 opour 1oAe rpn TA 1oddoo uo poqiospe ounjsrour 10 opour 1oAe rg n A uoryepnurio eArsreq PUL ION uoryezrpegs amp 1o Surlooo opeoseo xuej3 po ins snonuruoo YSU ureqo oS e1oA Ioquinu PUL UOISIOAUOO IOUIOUOUI JO OIJUOL uro qo1id Surunj 1o o1juo o1nj no snonurquoo Y ur oyerjsqns JOJIQIUI JO O1JUOD o3e1 poo uurm oo uorje nsrp reurq snonurjuo Stop pozrumdo Krjst WYP uorjsnquioo WOT SIDPIO 9 Q LIeA YPM SUOLA 9AT 1298U0 oyerjsqns ATOJIQIYUT PLM xmamo snonurjuo MOP xu 3 2rjeqerpe poso snonurquo MOP YURI eurieqgosr posojo snonurjuo punoAbs0q Oc cc CC Ch Ch Ch OOO OO Ch Ch Ch Ch Go Ch OC ri C CH CO CC CH ooo cco coo CC CC Ch Ch Ch Ch Ch Go Ch On C CH CO CC CH c P CN CO hi CN cO cCoO f CN cO cO bh aO N 0 9 GG qe 90 Fo 001 92 09 00r OV 0 8I 9 LLG VLE IST VLE 8y SOT D 09 9T OL OV
45. sisAjo oyd pue srsA o1pKu YILM uorjepeisop erqo ot Ar ojeiouoSop AIST o durexo oruropeoe snorio30u popon oje1ouoSgop A uSrq o durexo oruropeoe SNOLIOJON adoyost ue Jo amp e op oArjoeorpeq our amp zuo SurjeArjoeop UL opeoseo 1015691 ULSO Y ur 3sAqeqeo SurjeArjoeo q paq pues pezrpmg mot e ur oueqjourod0 Q9T T puno4bDsovq ooo Cc CH ooo uo ooo oo o ooo oo oO Ch Ch Ch Ch CH ooo CH NNN CH aN DO CC Ch cc CC CH CN iO a ON Co ooo Cc CH 10 10 CH oo OC Ch Ch Ch Ch CH N E Go Gi CY oO Co o N E GYG GVG GYG 84 84 v6 8cI GE 8I WI 89I 89I 8I 0c 9 001 0 9T 0 0 OV 0c 0c 06 6f 8I l 00 f NO C c ON ob mi m N qo aio lt y e Go ma CO EVIA cV1d IVa CUVd LSIG IH Vd LSIG JOSSIG CO LWTHSId LNWIdSId HVHquosid MOTISIA NIM SIG INIM SIG AdOId VIII LSIGAIG VIXAVIA SWLHg via ZNHHAHHG cdvuyuodd Tdvuaodd IN N4940 Nos d AVOA ZNALOVHd LOVHd DAANA IWDU 25 ponurquos a gS IS gS a uou uou IS SS gS gS 0S 0S 0S 9uou 9uou GIG DG 030p ozs ozs GET GET GET 80 80 uro qoid UOISNJO SUU ZUH 1032891 urKzu re nqnr uomnnos amp repunoq ds uonoe pue uorsngrt q 1039891 oryeurzuo snonuruoo ouirj ge JIM opour on un ooeurreud reour lopsu e1 S1eur oj qj1eo Jo orguoo Ted lopsu e1 SIBUI 0I 79 189 Jo orguoo peurd
46. soureuAp po dnoo jo moraryeq ILo uro qo1id onpe Arepunoq uro qo id usisop yond UOT PUI oyerysqns enq JoAIP 3Jeus ITULISUYJOSI UON SOUT SUITPOIIMS o qeLreA MOJ SOTPOQ u99A4 q uorjorg AIP OU 1OJVT IOSO SSeut oA J SOUIT SUITOIIMS o qeLreA OM sorpoq uooA3oq uorjorg AIP YPM ole Ipso SSVUI OM m Surqoqtas pordu sorpoq uooAjoq UOT AIP YIM IOje I2SO SSeUr oA T sdumf 399 117 euozejnqr amp uoud pue urrejre quourooe dsrp Snap eurdo ourt dun ouo euozejnqr amp uoud pue urrejre tquourooe dsrp Snap eurdo ourt Kd e1oq3ourouo 1ooueo 107 Surpnpouos Snap peurrd opy nsrp Au3ourrp JO uors1oAuoo orjA e 2 jUOISUVIT opyrnsrp Au3ourrp JO UOISIOAUOD orjAqe3e7 9 JOS eqep sorjoup uorjeroosse pojrur uorjeroossqT L 308 JEP sorjoup uoryerosse POPUT uorjeroossqT 9 99S eqep sorjoup uorjeroosse pojrur uorjeroossqT q jos eqep SOTJOUTY uorjeroosse pojrur uorjeroossqT y 398 ejep sorjoup uorjerosse Don uorjeroosstT puno4bDsovq ooo Ch on Go COCO vi CO CH ooo CC Ch CC CH ooo Ch Q my COO t CO CH ooo Ch CC Q lt H GO 00 GO GO lt Go GQ c CH SI q cq CO OQ TON 8C OT OT TS LG US OT sv 9 OV OV OV GE 88 99 GYG GVG GVG GVG GVG S o cO cO cO VD 00 00 c gt CH INAZNMW HH LZNH LI IdSZNW NOOZNG OYALNA CUIVINGHVMH TUVINGUVA OWVNAC Lond TV dq UAHA STHA AUC clu AUC IRIT AUC SITIO NA ISIGONUd HOS DAYA V Sana sa
47. t IR the time variable with 0 lt t lt T v JR the state variable of the coupled system of ordinary differential algebraic equations ut ui ui IR the state variables consisting of the partial differential variables ai IR 4 and partial algebraic variables ul IR and p IR is the parameter vector to be identified by the data fitting algorithm See also 2 89 for coupled ordinary differential equations 2 87 for flux functions 2 82 for algebraic equations and 2 70 for the most simple standard formulation Optionally the right hand side may depend also on a so called flux function f p ut ui x t where we omit for simplicity a possible dependency from coupled ordinary differential equa tions A solution depends on the spatial variable x the time variable t the parameter vector p the corresponding integration interval and is therefore written in the form v p t and ut p x t for i 1 Ma For both boundary points z and rg we allow Dirichlet or Neumann conditions of the 56 form ul p zp t ul p v t une p TR t up U t i 2 97 ul p EL t E p v t una p Zn t P p v t for 0 lt t lt t Again we do not require the evaluation of all boundary functions Instead a user may omit some of them depending on the structure of the dynamical model Note that boundary information is also contained in coupled ordinary differential algebraic equations Transition con
48. x 0 us p x 0 UNE D x 0 x 0 2 84 is violated after inserting Dirichlet or Neumann boundary values and corresponding approx imations for spatial derivatives then the corresponding system of nonlinear equations must be solved numerically proceeding from given initial values In other words consistent initial 46 values can be computed automatically where the given data serve as starting parameters for the nonlinear programming algorithm applied But even if we succeed to find consistent initial values for 2 84 by hand we have to take into account that the algebraic state variables and the spatial derivatives in the dynamical equation 2 84 are approximated numerically by the method of lines and suitable difference or any similar formulae The corresponding discretized equations of the DAE system are in general not consistent or more precisely are satisfied only within the given discretization accuracy Thus we have to assume that the resulting DAE system is an index 1 system unless it is guaranteed that consistent initial values for the discretized DAE are available see for example Caracotsios and Stewart 76 for a similar approach Example 2 18 HEAT A We consider again Example 2 17 now formulated as a first order PDAE up Du 2 85 0 v u with diffusion coefficient D The spatial variable x varies from 0 to 1 and the time variable is non negative i e t gt 0 The initial heat distribution for t 0 is
49. y AAA Figure 2 33 Non Continuous Transition Condition and Switching Point identify the system parameters the parameter defining the jump in the transition condition the switching time and the jump with respect to the time variable simultaneously within the accuracy provided by the experimental data However we have to be careful when defining the switching time Since the exact value T 0 25 is also an experimental time value we have to avoid that the least squares ob jective function becomes non differentiable at the optimal solution Thus the corresponding experimental data are omitted and some smaller lower and upper bound are defined for the optimization variable i e 0 21 lt z lt 0 29 64 2 6 7 Constraints In the previous sections we extended the general partial differential equation 2 70 step by step and got fitting criteria hy p t of the form hin t m Juin un p Tk t un p Tk t ust p Tk t v p t t gt 2 116 where u p p t denotes the solution of the partial differential algebraic equation 2 82 in the x th area see also 2 102 v p t is the solution of a coupled system of ordinary differen tial algebraic equations see 2 92 and z is the spatial variable value where measurements are available rounded to the closest line Thus the data fitting problem consists of minimizing one of the the objective functions Y Y uF s p amp ya 2 117 k 1 i 1 for the least
50. 0 RETURN BOUNDARY GRADIENTS 700 CONTINUE RETURN FITTING CRITERIA 800 CONTINUE FIT 1 V 1 RETURN CONSTRAINTS 900 CONTINUE RETURN GRADIENTS OF CONSTRAINTS 1000 CONTINUE RETURN GRADIENTS OF FLUX FUNCTIONS 42 C 1100 CONTINUE RETURN END Parameters to be estimated are L amd K Measurements are simulated subject to L 1 K 2 at the spatial coordinates 0 0 1 0 2 0 3 0 4 and 0 5 The corresponding input file is the following one where DFNLP is called for parameter estimation and RADA US for solving the discretized system of ordinary equations Starting values for DFNLP are L 2 and K 3 C NEASYFITNPROBLEMSNHEAT HEAT 6 Heat conduction One compartment Test 18 01 1995 deg mm min NPAR NPDE NPAE NCPL ICPL NCPB NRES NEQU NTIME NFIT IFIT NPLOT NOUT DQUPOI APRMET FLUX APRFLX 0 0 OPTMET OPTP1 100 OPTP2 OPTP3 OPTE1 OPTE2 OPTE3 ODEP1 ODEP2 ODEP3 k k O O O N F k O N 2 II OO OO GO tz OE 7 OE 1 0E 2 OO O ta N o 45 ODEP4 0 ODEE1 1 0E 5 ODEE2 1 0E 5 ODEES 1 0E 5 XSTART 0 0 COMP 1 1 0 11 1 1 0 L 0 0 2 0 10 0 K 0 0 3 0 10 0 SCALE 0 0 0 0 636619 100 0 0 1 0 237273 100 0 0 2 8 84335E 2 100 0 0 3 3 29598E 2 100 0 0 4 1 22844E 2 100 0 0 5 4 57849E 3 100 0 INTEG 0 ORDER 2 NLPIP 0 NLPMI 0 NLPAC 0 0 NBPC 0 DFNLP terminates after 22 steps wi
51. 0 0 13 13 0 0 0 0 0 0 14 14 0 0 0 0 0 0 15 15 17 47 1 48 73 1 16 16 0 0 0 0 0 0 17 17 0 0 0 0 0 0 18 18 0 0 0 0 0 0 19 19 0 0 0 0 0 0 20 20 10 8 1 48 17 1 21 21 0 0 0 0 0 0 22 22 0 0 0 0 0 0 23 23 0 0 0 0 0 0 24 24 0 0 0 0 0 0 25 25 5 81 1 41 76 1 26 30 3 23 1 27 32 1 27 40 0 93 1 20 7 1 28 50 0 32 1 11 18 1 29 60 0 1 1 5 76 1 Table 2 6 Experimental Data for Linear Kinetics Model 29 Linear Kinetics 50 O j 45 40 35 30 yo p t 25 15 10 Figure 2 15 Function and Data Plot for Constrained State Variable Example 2 13 DEGEN We consider a simple second order differential equation given by Bulirsch 63 y u y wu p sin pt 2 53 with initial values y 0 0 and y 0 p The equivalent first order system is Y Y n m 2 54 Ya Wu u t p sn pt If we try to integrate 2 53 or 2 54 by amy of the highly robust and efficient methods discussed in the previous chapter from 0 to 1 with u 50 and p 3 1415926535 we get the result displayed in Figure 2 16 Obviously the numerical solution fails because of internal instability of the equation even if we start with a very low integration accuracy say 10 19 To explain the instability we consider the general solution yi t sin pt esinh yt y2 t pcos pt eu cosh pt with e x p p Round off errors and slight numerical devia
52. 0 0 00T oor ANOO TII IS uomounj S1ouo ure1js orjso o S9tdozgjosr qiss rdurosu 0 Al TSZ 8 VIT AAH 0S sojyerjsqns pue souzuo eqoydorpAH 0 0 Ww S INAZNHGAH X UOTAN OS ZT ROT YA uorenbo om jo mag 0 0 G G CQ THIWNINIH A uorgjo zrp uoryeSedoad Jo uorjounj se Ajr5o o punos peurpny3uo 0 0 GI S VXHH uou o1oqdsruroq Jo JIABIS Jo 193u97 0 0 DOT HdSIWHH x uonnjos yoexo s1ojourered quejsuoo YIM UOISNYIp re 9urr 0 0 6 zZ XX LVAH d pnis LSIN toddoo jo uorsuedxo emmy 0 0 98 A INHVH ouou 78 suoryenbo e jo ura3s s uoryezturido eqo 8 10 uro qo1d 3s9T T li I I 905 ouou 78 suoryenbo e jo tua3s s uorjezrumdo eqo s 10 uro qo1d 3s9 0 0 e I GOD uou ZSE suoryenbo e jo uro3sAs uoryezrurado eqo 8 10 uro qo1id 3s9 e I e FOOD uou 78 suoryenbo e jo uro3sAs uorjezrumdo peqo 8 10 uro qo1d 3s9 0 0 e e CO quel 28 uoryezrurado eqo 8 10 uo qo1d ent 0 0 qI G cOD supu G8E uorjezrumdo eqo 8 oj uro qo d ent 0 0 qI G TOO A 981 IOAOUIN oSOOn S OAIA U 0 0 er Tt ALVI ATS 020p fas punobyopqg w wu u EH Dono wd 10 LO on ZOO OD DD D D iO nr LO Hx mm gS gS SS SS 030p Fc 96 Tac 608 LIZ LIZ LIG foa uorjoung zj1odurior Aq ayer Ay ey Ion uorjenbo ornou XN9SULY O UBUUL M pouo gq 1039891 uorjezriourAq od Y ptsur 1039891 uorjezriourAq od ptsur 1039891 uorjezriourAq od ptsur 1039891 uorjezriourAq od e ptsur Xpn3s LSIN uondiospe Apnys LSIN
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54. 08 08 08 08 6I 6I SET SET OV 207 ve GT 08 8 SOG Y Y N CH CN CN CN CN CN CN DM Y 00 g Ow AN N D s OV CN ND CO Y F MOOHS X VIVOS Vuvos GIE V HdSOHd WOTOACNdd dd AUNAd AT ONT ST NGNdd GT ONT IT NGNdd SO NGNdd SLNWHGId GLNWHGI d LLNSHGI d LNWGI d HIA WAN HVIGUVIN Ud NIJI NAGOHGAH HO LVHOXM YOJVAH eTILLSIG TILLSIC SAS VG X GNT VG GNI AVG IWDU 43 OS SS dd dd SS IS IS 030p 6c 6c c 6T Pma eorrjoopo uoryenbo oq 19p UBA 8jos ggep eguourLrodxo 99 119 tw uorgd rospe re noseA uorjdaospe repose USISOp eguourtiodxo 1030891 qo3eq rums Y UT uoreor ULYPIN SIYSIOM YM USISOp ejyu umri odx 1030891 YO eq mues e ur uorjoeol ULYPIN S OSS9A poop OM YIM 1032991 YOJeVq TULOS Y UI UOTJOVOI ULYPAIN Te Sur ooo UJLM 1049891 repnqny Kreuorje g uro3s amp s Kpoqrj nur popou xona Surjye roso AIST togr dure 103strsue SOTJOULY u ju x st 9wuor Aq aye3sqns Jo y1odsue punosbyonq N c DO ri MN c CN co C Du CO N O m Pus OT ve VI OV FOI 0 42 v8 LI cOT l I qq GG e co co oi TOddA SSUV SVA dS V SVA XNVHLWUn ANNVHLWHO NVH LWHO UV TINE AL MONAL LSISNVAL av NVUL IWDU 44 10 6 Partial Differential Equations Now we proceed from r data sets ti yf t lye cyl kemi eT where l time values and l lr corresponding measurement values are defined Mo
55. 1 generation of output files 25 a6 4x i5 DQUPOI Number of points used to approximate the spatial derivatives for method of lines by polynomials of de gree DQUPOI 1 DQUPOI must be odd and at least three for polynomial interpolation 30 a6 4x 15 APRMET Determines how to approximate ET S NN derivatives APRMET 0 Polynomial approximation for uz us in case of FLFLAG 1 For FLFLAG gt 1 APRMET 1 central differences for u and recursive application for uz DQUPOI gt 2 APRMET 2 5 point differences for u and recursive application for uz DQUPOI gt 4 APRMET 3 5 point differences for uy and separately for Up APRMET 4 Forward differences for u APRMET 5 Backward differences for uz APRMET 6 Individual selection of discretization formula for uz can be different for each state variable a6 4x 15 FLFLAG Indicates existence and type of flux function FLFLAG 0 No flux function defined FLFLAG 1 Flux function exists and is differentiated by chain rule FLFLAG 2 Flux function exists and is discretized by the method defined below a6 4x 15 APFLUX Parameter for choosing high resolution scheme If 1 lt APFLUX lt 1 the one parameter family of TVD schemes of Clhakasstihy and Osher is used Al APFLUX 1 0 upwind scheme APFLUX 1 3 no name APFLUX 0 0 Fromm scheme APFLUX 1 3 third order upwind biased scheme APFLUX 1 2 second order scheme APFLUX 1 0 central difference scheme A
56. 1 0 50 0 1000 0 200 4 1 0 60 0 1000 0 209 3 1 0 90 0 1000 0 202 4 1 0 105 0 1000 0 213 1 1 0 1 0 5000 0 72 9 1 0 3 0 5000 0 153 8 1 0 5 0 5000 0 189 4 1 0 10 0 5000 0 239 7 1 0 15 0 5000 0 250 6 1 0 20 0 5000 0 234 1 1 0 25 0 5000 0 255 1 1 0 30 0 5000 0 254 7 1 0 35 0 5000 0 266 2 1 0 40 0 5000 0 270 3 1 0 50 0 5000 0 0 0 0 0 60 0 5000 0 0 0 0 0 90 0 5000 0 237 7 1 0 105 0 5000 0 250 8 1 0 NLPIP 0 NLPMI 0 NLPAC 0 0 NBPC 0 A solution is obtained by DFNLP in 19 iterations see Figure 9 2 The Fortran code for the function and gradient evaluation is defined as follows Q Q 10 SUBROUTINE SYSFUN NP MAXP NO MAXO NF MAXF NR MAXR X Y T C YP YO FIT G DYP DYO DFIT DG IFLAG IMPLICIT DOUBLE PRECISION A H 0 Z DOUBLE PRECISION K1 K2 DIMENSION X MAXP Y MAXO YO MAXO YP MAXO G MAXR FIT MAXF DYO MAXO MAXP DYP MAXO MAXP MAXO DG MAXR MAXP DFIT MAXF MAXP MAXO BRANCH W R T IFLAG IF IFLAG EQ 0 RETURN K1 X 1 K2 X 2 R X 3 GOTO 100 200 300 400 500 600 700 800 900 IFLAG RIGHT HAND SIDE OF ODE 0 CONTINUE 15 Q Q Q Q Q Q Q YP 1 K1 R Y 1 C Y 1 K2 Y 1 RETURN INITIAL VALUES FOR ODE 200 CONTINUE Y0 1 0 0 RETURN FITTING CRITERIA 300 CONTINUE FIT 1 Y 1 RETURN CONSTRAINTS 400 CONTINUE RETURN GRADIENTS OF RIGHT HAND SIDE OF ODE W R T X AND Y 500 CONTINUE DYP 1 1 R Y 1 C Y 1 DYP 1
57. 1 with a termination tolerance of 1079 After 60 iterations the termination conditions are satisfied at kis 0 0984 and ka 0 0512 Function and data plots are shown in Figures 2 5 and 2 6 It is even possible that break points become variables that are to be adapted during the optimization process However there is a dangerous situation if during an optimization run a variable switching point passes or approximates an experimental time value If both 18 Linear Compartment Model with Switching Points 60 50 40 ya p t 30 20 10 Figure 2 6 Function and Data Plot for Compartment 2 coincide and if there is a non continuous transition then the underlying model function is no longer differentiable subject to the parameter to be optimized Possible reactions of the least squares algorithm are slow final convergence rates or break down because of internal numerical difficulties On the other hand variable switching points are highly valuable when trying to model for instance the input feed of chemical or biological processes given by a bang bang control function or any other one with variable break points To give an example consider a pharmacokinetic model with an initial lag time that is unknown a priori Example 2 8 LKIN_LA We consider the same linear compartment model as before but now with a lag time T f 0 M iem yi d yi 0 Do ban d T
58. 109 On see Timoshenko and Goodier 507 where the load function f is given by f x t art ast 3 z 37 E 0 01 is the constant for flexural frigidity and k a4 and az are parameters we want to identify The spatial variable x varies between 0 and 3 where the transition is defined at xi 1 2 Initial conditions are w p 1 0 Wee p x 0 0 and also the boundary conditions are set to zero i e p 0 w p 9 t Weep 0 0 Uss m 9 4 0 The joint condition requires that the solution values w p x t and the third derivatives iss D 3 0 coincide at z z Experimental data are simulated at 10 time values 0 2 0 4 2 0 and 9 spatial values 0 3 0 4 2 7 perturbed subject to an error of 5 Z Exact starting and final parameter values are shown in Table 2 14 together with 5 confidence intervals The two integration areas are discretized with respect to 21 and 25 lines and a five point difference formula for first and second derivatives The integration is performed with termination tolerance 1075 DFNLP computes the solution after 10 iterations with final accuracy 10 Figure 2 32 shows the final state variable w 2 6 6 Switching Points We consider now the same model that was developed so far in the previous sections i e a system of one dimensional partial differential algebraic equations with flux functions coupled ordinary differential equations and different integration areas with trans
59. 2 43 Ya k12Y1 baus y2 0 0 with an initial dose Dg 100 Experimental data are simulated under the same conditions as for Example 2 7 but now with T 5 as additional model parameter to be estimated and without further switching points DFNLP is started at T 1 ki 1 and ko 1 with a termination tolerance of 10719 After 59 iterations the termination conditions are satisfied at T 5 21 kj 0 1022 and ks 0 0499 Function and data plots are shown in Figures 2 7 and 2 8 So far we considered only given switching points which are stated explicitly as part of the model formulation However there are very many other reasons for discontinuities of 19 Linear Compartment Model with Lagtime 100 90 80 70 60 y p t 50 40 30 20 10 o NN Aa a e 00 e 100 Figure 2 7 Function and Data Plot for Compartment 1 Linear Compartment Model with Lagtime 60 50 40 ya p t 30 20 10 o L e S D e 00 e 100 Figure 2 8 Function and Data Plot for Compartment 2 the right hand side of a system of ordinary differential equations where switching points are defined implicitly They appear especially in chemical engineering or multibody systems see Preston and Berzins 390 in the first and Eich Soellner and F hrer 132 in the second case A typical ex
60. 337 342 26 27 28 29 32 33 34 35 36 38 39 40 41 Bard Y 1970 Conparison of gradient methods for the solution of nonlinear parameter estimation problems SIAM Journal on Numerical Analysis Vol 7 157 186 Bard Y 1974 Nonlinear Parameter Estimation Academic Press New York London Bartholomew Biggs M C 1995 Implementing and using a FORTRAN 90 version of a subroutine for non linear least squares calculations Report NOC Hatfield Bauer I Bock H G K rkel S Schl der J 2000 Numerical methods for optimum exper imental design in DAE systems Journal of Computational and Applied Mathematics Vol 120 1 25 Baumeister J 1987 Stable Solution of Inverse Problems Vieweg Braunschweig Bazeze A Bruch J C Sloss J M 1999 Numerical solution of the optimal boundary control of transverse vibrations of a beam Numerical Methods for Partial Differential Equations Vol 15 No 5 558 568 Beck J V Arnold K J 1977 Parameter Estimation in Engineering and Science John Wiley New York Bellman R E Kalaba R E Lockett J 1966 Numerical Inversion of the Laplace Transform American Elsevier New York Belohlav Z Zamostny P Kluson P Volf J 1997 Application of a random search al gorithm for regression analysis of catalytic hydrogenizations Canadian Journal of Chemical Engineering Vol 75 735 742 Beltrami E 1987 Mathematics for Dynamic Modeli
61. 42 State Variable x2 t Nonlinear Kinetics of a 3 Blocker 60 50 40 20 10 Figure 2 43 Computed and Exact Control Variables for n 10 20 40 short steps of the ODE solver As mentioned above these switching points may become optimization parameters A special situation arises if we consider bang bang controls where control function values jump from one constant level to another one By the next example we illustrate the usage of a bang bang control the approximation of a boundary function and the possibility to minimize also the final integration time in case of a partial differential state equation Example 2 29 TIME_OPT We consider now another variant of our standard test prob lem Example 2 17 the heat diffusion model On the one hand we want to approximate a given final boundary function f z att T as closely as possible on the other the final time T is to become as small as possible Thus the problem is a mixture of a minimum norm and a time optimal one see Schittkowski 424 A constant scaling parameter a is introduced to weight the two different objectives and we get the objective function 1 J T 5 u T s x T J z 2dz oT 2 135 0 subject to the state equation ui T s d Mes u T eU d 0 2 136 u T s 0 Se i u T s 1 t uz T s 1 t s For our numerical test we choose a 0 01 and f x 0 5 U
62. 5u 49 Molecule Diffusion 500 Figure 2 26 Diffusion Variable Defined by Flux Function and ut falu Din gt see Thomas 503 In this case the exact solution is known 1 u p x t 5 a a and can be used to simulate 54 measurement data at t 0 1 t 0 9 and xz 0 1 x 0 6 subject to p 0 01 Subsequently these data are perturbed with an error of 5 96 The exact solution is used to generate initial values and Dirichlet boundary conditions The partial differential equation is discretized by 51 lines and a second order upwind scheme The least squares code DFNLP is started at p 1 together with an implicit solver for the system of 49 ordinary differential equations After 39 iterations we get the estimated parameter p 0 01031 The resulting surface plot is shown in Figure 2 27 We see that the viscous i e the parabolic part of the equation smoothes the edges 2 6 4 Coupled Ordinary Differential Algebraic Equations A particular advantage of applying the method of lines for discretizing a partial differential equation is the possibility to couple additional ordinary differential algebraic equations to 50 Burgers Equation RIA AS em PS A Figure 2 27 Solution of a Hyperbolic Equation the given partial ones We proceed from the general explicit formulation Oua Fy p U Ug Ugg U t t l 2 89 0 Pel DOs sz Utt gt co
63. 9 325 336 Bischof C Carle A Corliss G Griewank A Hovland P 1992 ADIFOR Generating derivative codes from Fortran programs Scientific Programming Vol 1 No 1 11 29 Bitterlich S Knabner P 2003 Experimental design for outflow experiments based on a multi level identification method for material laws Inverse Problems Vol 19 1011 1030 Bjorck A 1990 Least Squares Methods Elsevier Amsterdam Black F Scholes M 1973 The pricing of options and corporate liabilities Journal of Political Economics Vol 81 637 659 Blatt M Schittkowski K 2000 Optimal control of one dimensional partial differential algebraic equations with applications Annals of Operations Research Vol 98 45 64 Blom J G Zegeling P A 1994 Algorithm 731 A moving grid interface for systems of one dimensional time dependent partial differential equations ACM Transactions on Mathe matical Software Vol 20 No 2 194 214 Bock H G 1978 Numerical solution of nonlinear multipoint boundary value problems with applications to optimal control Zeitschrift fir Angewandte Mathematik und Mechanik Vol 58 407 Bock H G 1983 Recent advantages in parameter identification techniques for ODE Pro ceedings of the International Workshop on Numerical Treatment of Inverse Problems in Dif ferential and Integral Equations Birkhauser Boston Basel Berlin 95 121 Bock H G Eich E Schl der J P 1987 Numerical solution of constrained
64. Corresponding Fortran source code only complete ver sion Frame of a Fortran code for estimating parameters in explicit model functions Frame of a Fortran code for estimating parameters in differential equations Frame of a Fortran code for estimating parameters in differential algebraic equations Frame of a Fortran code for estimating parameters in steady state systems Include file with dimensioning parameters for MODFIT Solving parameter estimation problems in systems of one dimensional partial differential equations and par tial differential algebraic equations Corresponding Fortran source code only complete ver sion Frame of a Fortran code for estimating parameters in systems of partial differential equations Include file with dimensioning parameters for PDEFIT DOS batch file to execute the Fortran compiler can be modified interactively DOS batch file to execute the Fortran linker can be modified interactively i Plot programs and editor SP PLOT EXE Standard plot program where input data are read from files GNUPLOT EXE Public domain plot program Gnuplot EDITOR EXE Syntax highlighting external editor Database EASY FIT MDE Main database of EAS Y FIT VodelDesigr containing ta bles forms reports macros and modules EASY FIT HLP Corresponding help file EASY BIT CO Icon file for EASY FTTModelDesign EASY FIT PDF Adobe Acrobat Reader file containing complete docu mentation Subdirectories OBJECT Dire
65. IUO YIM 10J9891OIQ qQojeq po q pezrpeuriou 21921 AY your jo uorso dxo Tal SUOGILIOIPAY SNOTILA 0 OULTJOUI JO uors19AUO ooepms ptro e39 A Tepour orureu p eorurouootqq uorneourred oueriquiour Aq uorje redos ser 1032891 9U IQqUIOUI UOI2U9391 JPO Surdurep pue Ses Arorgse o Yy urojsKs Surje so eorueqoopA poe jndur jo uSrsop uumn oo uore qsrp JUoMOdUTIOOTNUL snonurquo uumjoo uorjej srp 3ueuoduroorj nur snonurquo uorje ndod sunen uorjepnoaro oyADOYCurAT ounuruiq uorjenbo erjuod9grp VIIy oA ex90 T uorjenbo erjuod9grp L193 OA ex90 T Sutjejrso AIS uoryenbo zuo10 q uorjenbo zuodo q IS9AIBY juejsuoo UA Q1AOIS I1YSTSO T euiqmmbo jo uorgenurquoo pue uoreoo q 10 119 uorjeumxo dde 5 10 opour quouryreduroo euy dutt suorjenbo amp 31Arjrsuos TIM popou 3uourj reduroo reour dur Sjure1jsuoo orureu p opour juourj reduroo Tout o duutg sosop oo1u3 YIM opour 3uourj reduroo mom o duitg SOATYRALIOP jror dxo opour juourjTeduroo resu o duitg SUIT Ser o qerreA ILM opour 3uourjreduroo reou o duug 193 oureded quepunpor Tre u ouo Topour juourj reduroo reou o duitg punosbyonq ooo Ch CC Ch Ch CH T OG CC Ch CH OC Ch Ch Ch Ch Ch O Ch Ch CH 8 0 0 G 0 0 G 0 8 0 9 0 0 v 0 06 0 06 0 8 0 GI 0 G 0 G 0 0 0 I 0 G 0 G 0 8 IG lt 0 G 0 8 0 G 0 G tu u 0 0 0 8y Ur 0891 9 04 OES OTI 001 091 06 0 00 OVE 9
66. In case of numerical approximation of gradients the differential equation must be solved as accurately as possible Again it is recommended to start with a relatively large accuracy say 1 0E 6 and a low number of iterations and to increase the accuracy when approaching a solution by restarts Initial Stepsize for Solving Differential Equations Define initial stepsize for differential equa tion method used The parameter is adapted rapidly by internal steplength calculation Bandwidth of Jacobian of Right Hand Side Implicit methods require the evaluation of the Jacobian of the right hand side of an ordinary differential equation with respect to the system parameters since they have to apply Newtons method to solve certain systems of nonlinear equations The Jacobian is evaluated either numerically by the automatic differentiation features of PCOMP or must be provided by the user in form of Fortran statements In any case is possible that the Jacobian possesses a band structure depending on the ODE sys tem EASY FIT Desion allows to define the bandwidth that is passed to the numerical integration routines to solve systems of internal nonlinear equations more efficiently The bandwidth is the maximum number of non zero entries below and above a diagonal entry of the Jacobian and must be smaller than the number of differential equations When insert ing a zero value it is assumed that there is no band structure at all The bandwidth can be
67. N 1963 Deterministic nonperiodic flow Journal of Atmospheric Sciences Vol 20 130 141 Loth H Schreiner T Wolf M Schittkowski K Schafer U 2001 Fitting drug dissolution measurements of immediate release solid dosage forms by numerical solution of differential equations submitted for publication Lubich C 1993 Integration of stiff mechanical systems by Runge Kutta methods ZAMP Vol 44 1022 1053 Lucht W Debrabant K 1996 Models of quasi linear PDAEs with convection Report Dept of Mathematics and Computer Science University of Halle Germany Lucht W Strehmel K 1998 Discretization based indices for semilinear partial differential algebraic equations Applied Numerical Mathematics Vol 28 371 386 Luenberger D G 1979 Introduction to Dynamic Systems Theory Models and Applica tions John Wiley New York 20 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 Luksan L 1985 An implementation of recursive quadratic programming variable metric methods for linearly constrained nonlinear minmaz approximations Kybernetika Vol 21 22 40 Luus R 1974 Two pass method for handling diffcult equality constraints in optimization AIChE Journal Vol 20 608 610 Luus R 1993 Optimization of fed batch fermentors by iterative dynamic programming Biotechnology and Bioengineering Vol 41 599 602 Luus R 1993 Optimal control
68. New York Jennings L S Fisher M E Teo K L Goh C J 1990 MISER3 Optimal Control Software Theory and User Manuel National Library of Australia Jiang G S Levy D Lin C T Osher S Tadmor E 1997 High resolution non oscillatory schemes with non staggared grids for hyperbolic conservation laws UCLA CAM Report 97 7 Dept of Mathematics University of California at Los Angeles Jiang G S Shu C W 1995 Efficient implementation of weighted ENO methods UCLA CAM Report 95 42 Dept of Mathematics University of California at Los Angeles Johnson C 1998 Adaptive finite element methods for conservation laws in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations B Cockburn C Johnson C W Shu E Tadmor eds Lecture Notes in Mathematics Vol 1697 Springer Berlin Johnson R C Jasik H 1984 Antenna Engineering McGraw Hill New York Johnson R S 1970 A nonlinear equation incorporating damping and dispersion Journal of Fluid Dynamics Vol 42 49 60 16 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 Jourdan M 1997 Simulation und Parameteridentifikation von Destillationskolonnen Diploma Thesis Dept of Mathematics University of Bayreuth Germany Juedes D W 1991 A taxonomy of automatic differentiation tools in Automatic Differ entiation of Algorithms Theory Implementation and Application A Griew
69. OF o0 iO co O HN o N O MAN Qo CN CN NDN cx aO 10 N aO CO CN Y SNNN a NIMOG SODINWINIOG OND NOJYULSO OAL ULSO LAG HLSO ULD ULSO TLO ULso INS LLSO ULSO ANHOLSO quor Lso OVA SO ANVUO AOTT HO cddddOoO q Y4ddO9D ddddOD IHO IOOD TOOD UAANOD NALNODO NOO LNOO ILLSNOO VHAYSNODO SIHNINOO O IHNOO COTANOD IWDU 24 ponurjuoo GI IS gS SS SS SS SS gS gS 030p vez s97 sec vez roc Je 931 pez 971 fo JOS eJep SOIJOUTY uorjeroosse Dom uoryeroosst T T 398 VCP SOIOUTY uoryeroosse Dom uorjeroosst T T 39s Jep SOTJOUTY uoryeroosse POPUJ WOTYRIOOSSTCT THVALSIA Woy eyep urags s Jojourered AGO peanqruagsiq Domm I9J9p19AO pue o qejsun AIS urojs s 1ojourered pomqys q SOTJOULY WOTYRIOOSST T uors1odsrp erxe YM 1039891 re nqni eurr 9q3osr uoN oAIno quourooe dst T ojeuoq eo WNT q3tr4 I9PIOSTP oArsso1dop orueur SUTYeALT oseyd prnbr 1039891 PLWY V ur so3e1jsqns JO uornqr3st T z PS eyep 3uourLiodxo eorynooeurreud e Jo stat 3uourooe dst T gos eyep 3uourriodxo eorynooeurreud e jo sonour 3uourooe dsiT IOj9 D2sO oporp Touun J xo d uioo Iouirp ouo pue soouejsqns OM UA TOPOUL orjoup oxreurreu Y uonerpmsrp perjuodogrp Tacho my oqodorur JO YYMOIS orxnerTq juourogeueuir sojoqer q Auoydip 03 ouozuoq Jo uoryeuoSo1p qop I q uors I9A e1ed stsAjojoyd pue stsATorpAY yya uorepe19 p erqo dorpN
70. U when calling MAXPDE the subroutine with IFLAG 11 FLUXUX MAXPDE Has to get the derivatives of the flux functions for UX when calling MAXPDE the subroutine with IFLAG 11 FLUXP MAXPDE Has to get the derivatives of the flux functions for PAR when calling the subroutine with IFLAG 11 If INTEG is set to 1 PDEFIT computes the integral J u p x t dx 954 where j 1 n andi 1 ny The integral is evaluated by Simpson s rule and denoted by SIMPSN I J in the PCOMP language In case of a Fortran implementation of the model functions the same value can be retrieved from a public common block kept in the include file PDE FIT INC under the name AINTEG I J Note that also access to the complete solution array u p p t at the discretization points zk is also possible The corresponding array is denoted by USOL LK The time value is either a measurement value or an intermediate value needed for generating plot data Example 9 5 HEAT We consider now Example 6 6 again a simple heat conduction model found in Schiesser 420 where Fourier s first law for heat conduction leads to the equation te Usa 9 5 defined for 0 lt t lt 0 5 and 0 lt z lt 1 Boundary conditions are u 0 2 u 1 t 0 9 6 for 0 t 0 5 and the initial values are u z 0 sin 9 7 40 for 0 lt z 1 and 0 lt L 1 In addition we are interested in the total amount of heat at the surfa
71. UL opouir OTSeTY SUIpMoID pue As1d 10yepald yuoyxquerd yya KSo ooo uoryerudoq pigos e jo Sur amp a q ULIO uorjoeor juejsuoo uou pue uorsiodsrp erxe JIM 1039891 re nqnj eurreq30st uON uors1odsrp erxe TIM 1099891 re nqn1 RULIOY OSI UON AOT 9AIJ99APR A1S19dsI GT 10jsoSIp orqo1o ue poq poxiq S1ojoure red quejsuoo YIM uorjenbo Gott aseyd pros ur uorgoeor pue uorsngrt q AGO p ldnoo eseyd pros ur uorjoeor pue Dog xnp snonurquoo suro3s amp s eorgo orq ur Suruorjrred pue Gott uornisuer xng snonurjuoo surojs s eorgo orq ur Suruorgn red pue Gott uornisuer snonurquoo uou surojs s eorgo orq ur Suruorred pue gott Son eA Terur erjyuouodxo suro3s amp s eorgo orq ur Suruorr red pue uorsngiGq uornisuer snonurjuoo uou surojsAs eorgo orq ur Suruorryred pue uorngiGq Tes oqny YSno1yy uorsngtp YIM MOTT uonipuoo repunoq reourpuou YIM uorsngrp 1Ie ur uoN 19j A ut Togo JO uorsngrt q sjuororgeoo snonurquoostp YPM uro qo1d uorjooAuoo uorsngrt q uorjoeor uorjd rosqe pue Gott suon Ipuoo Krepunoq uueumoN pue yopuq yya werqoid ost sjos ejep seore Z YIM oueiquiour sms erp ugnorqj uorsngip ojeizjsqng punosbyo0q ooo CH S SO OC 10 O OO oo cocco NOOO O ei CO CO Ch Ch CO CH CO CO CC Cu CH 200 ri Cl CN CY rd ri r rd rd ri ond CI N ri ri rd rd 0c 0c 09 Ur GL 0 GG T I8 C66 C66 0 0c OT 0 OT cl GE 0G OV 0G ISI 86 NN Co Y
72. a collection of 12 real life case studies Most of the case studies possess an industrial background The software system is implemented in form of a database under Microsoft Office Access 2007 running under Windows XP or higher and comes with the royalty free runtime version The underlying numerical algorithms are coded in Fortran and are executable independently from the interface Model functions are either interpreted and evaluated symbolically by a program called PCOMP permitting automatic differentiation of nonlinear model functions or by user provided Fortran subroutines In the latter case interfaces for the Fortran compil ers Watcom F77 386 Salford FTN77 Lahey F77L EM 32 Compaq Visual Fortran Absoft Pro Fortran Microsoft Fortran PowerStation and Intel Visual Fortran for Windows 32 and Windows 64 environments are provided Important Notes 1 Trademarks Windows Microsoft PowerStation are registered trademarks of Microsoft Corp WATCOM is a registered trademark of WATCOM Systems Inc FTN77 is a trademark of Salford Software Ltd INTEL is a trademark of Intel Corporation Adobe Acrobat are registered trademarks of Adobe Systems Inc 2 Copyrights GNUPLOT Copyright 1986 1993 1998 2004 Thomas Williams Colin Kelley RADAUS DOPRI5 Copyright 2004 Ernst Hairer LAPACK Copyright 1992 2007 The University of Tennessee 3 Data Fitting Codes Note that with Version 4 32 of EASY FTT ModelDesign the data fitting algorithm DF
73. a restriction is to be formulated Many practical control problems are given in form of higher order differential equations with boundary conditions However arbitrary nonlinear equality constraints with respect to the optimization parameters can be added to the optimal control problem Thus an equivalent optimal control problem with boundary values is easily formulated see Section 2 4 6 and especially Example 2 15 Example 2 28 B BLOCK By the first example we consider the optimal control of an ordinary differential equation where we know the exact control function The goal is to find an optimal distribution of a drug more precisely a B blocker so that a given concentration level is followed as closely as possible see Cherruault 90 The underlying pharmaceutical system is modeled by two kinetic differential equations Ly kyp km kaum s t Ti 0 H 2 134 Ta k1201 koi Xo x2 0 Initial substrate concentration is a 18 The transfer coefficients could have been obtained by a previous data fitting run see Section 2 4 1 and are given by kis 2 9545 ko 5 7214 and ke 0 3658 see Cherruault 90 Control function s t is piecewise constant with switching points 1 n for n 10 n 20 and n 40 respectively and function values 1 Sn The distance of x1 t from the given goal function f t a is evaluated at 40 equidistant points between 0 and 1 The numerical solution of the ODE is restarted at
74. are always started at t 0 In the first case shifted time values are different from the original formulation and somehow misleading in the second case we get zero solution values from t 0 to t to that are out of interest for the user The second case allows to treat to as a variable initial time to be estimated if we declare it as an optimization parameter However we are supposing that in most practical situations the initial time is indeed zero To become a bit more flexible in case of non zero initial times EASY FIT 4 lt 9 allows to define negative experimental time values If the first measurement time t is negative then the integration is started at t t lt 0 with initial value y p ty c yo p c which may depend also on the parameters to be estimated and the concentration parameter A possible application is found in the example Example 2 16 PHA_DYN1 2 3 We consider a pharmacodynamic process of the form Ly k 3 A La k o Ta 23 8o m 23 k F kP ro 2 68 T3 kb o Lg z3 c 23 ki with constants sy 1 zo 0 5 and Kin 10 and c is a concentration parameter kp ki kb and k are parameters to be estimated with initial values kp 2 k 5 H 3 and ky 10 Experimental data are shown in Table 2 8 for five different concentration values c 0 005 c 0 05 c 0 25 cs 0 5 c4 0 75 and cs 1 First we observe that experimental data are only availab
75. artificial weight of 1 0E 7 is inserted 5 There are no break points Integration is performed only from one shooting point to the next and then initialized with a shooting variable The differences of shooting variables and solution at right end of previ ous shooting interval lead to additional nonlinear equality constraints The shooting index determines the number and position of shooting points O no shooting at all 1 each measurement time is a shooting point 2 every second measurement time is shooting point 3 every third measurement time is shooting point etc Method for Solving Ordinary Differential Equations The numerical parameter estimation code MODFIT possesses interfaces to two subroutines for solving ordinary differential equa tions 21 1 Implicit Radau type Runge Kutta code RADAUS Copyright 2004 Ernst Hairer of order 5 for stiff equations confer Hairer and Wan ner 199 2 Runge Kutta Fehlberg method of order 4 to 5 see Shampine and Watts 470 with additional sensitivity analysis implemented by Benecke 36 ODE Model Parameters E xj Number of Differential Equations 2 E Number of Measurement Sets 2 8 Break Paints lt M A Number of Concentration Values 0 4 C nstisinte Shooting Index 0 E e implicit C explicit A Final Accuracy absolute 1E 06 J Initial Stepsize 0 0001 4 Integration Method A A Final Accuracy relative 1E 06 2 Bandwi
76. become unstable especially if non continuous boundary conditions are supplied to describe the propagation of shocks see Schiesser 120 for some numerical examples The following upwind formulae are available for solving hyperbolic equations simple upwind formula second order TVD scheme third order upwind biased TVD scheme For more information see the original literature e g Yee 557 Chakravarthy and Os her 82 83 84 Sweby 500 Wang and Richards 541 and Yang and Przekwas 556 TVD stands for total variation diminishing and the corresponding one parameter family of upwind formulae was proposed by Chakravarthy and Osher 82 In this case a certain sta bility criterion requires that the internal time stepsizes of the ODE solver do not become too small compared to the spatial discretization accuracy Because of the black box approach used the stepsizes however cannot be modified and we have to suppose that the criterion remains satisfied c ENO Method for Systems of Advection Equations Systems of non homogeneous nonlinear advection equations fi uk fa p ul ub z t 4 4 with area index i i 1 n with u IR n gt 1 can be solved by essen tially non oscillatory ENO schemes see Harten Engquist Osher and Chakravarthy 206 Harten 207 or Walsteijn 536 High order polynomials are applied to approximate a so called primitive function which is supposed to represent the flux f
77. between zero and the maximum experimental time value Time values are ordered internally If the table contains no entries at all it is assumed that there are no constant break points with respect to the time variable On the other hand it is possible that the last n optimization variables to be estimated are used as break points in the code that defines the right hand side and the initial conditions In this case the number n must be inserted If n gt 0 and constant break points exist in the corresponding table then these values are ignored 38 Constraints Restrictions are allowed for models based on partial differential equations and can be formulated in form of equality and inequality constraints with respect to the pa rameters to be optimized and the solution of the dynamical system at some of the given experimental time and spatial parameter values Where the total number of constraints can be retrieved from the subsequent table the number of equality constraints must be supplied Equality restrictions must be defined first in the input file for model functions The table allows to define time and spatial values for which solution values of the underlying dynamical system can be inserted Order Serial order number of constraints Equality constraints must be defined first Name Arbitrary name for the constraint to be printed in reports Time Corresponding time value at which a constraint is to be evaluated Note that the
78. both types of boundary conditions Discretization In case of addressing individual difference formulae to the state variables the following approximation schemes can be combined 1 3 point difference formula recursively for second derivatives 2 5 point difference formula recursively for second derivatives 3 forward differences for first derivatives 4 backward differences for first derivatives Spatial Positions of Coupled ODE s The positions of the spatial variable x where ordinary differential equations are coupled to the system of partial equations must be given The order must be increasing and any decimal value within the integration interval is allowed The number and order of entries must coincide with the number and order of ODE s defined in the model function file Decimal numbers are rounded to the nearest integer that describes a line of the discretized system Spatial Positions of Coupled Algebraic Equations The positions of the spatial variable x where algebraic equations are coupled to the system of partial equations must be given The order must be increasing and any decimal value within the integration interval is allowed The number and order of entries must coincide with the number and order of algebraic equations defined in the model function file Decimal numbers are rounded to the nearest integer that describes a line of the discretized system Spatial Positions of Fitting Criteria The positions of the spatial var
79. can be applied to compute consistent initial values The equations are added to the algebraic partial differential equa tions and the whole system of nonlinear equations must be solved simultaneously Algebraic differential equations are highly useful in case of implicit boundary conditions since the spatial coupling values may coincide with boundary points Each algebraic equation requires also the declaration of a corresponding algebraic variable for instance for the partial state variable or its derivative at a boundary point Thus one has do define also some trivial Dirichlet or Neumann conditions containing only the algebraic variable at the right hand side 53 Diffusion of Drug in a Saline 600 000 500 000 400 000 300 000 200 000 100 000 Function and Data Plot Figure 2 28 Diffusion of Drug in a Saline State Variable Figure 2 29 54 Example 2 22 HEAT_NLB We consider a simple parabolic equation see also Tr ltzsch 510 Ut Yeu with initial value y x 0 cos x a homogeneous left Neumann boundary condition y 0 t 0 and another implicitly defined boundary condition u t e t y zn t ue zn t 0 at the right end point rg 0 785398 We have exp t exp 1 3 e t 0 25 exp 4t exp 2 3 exp 1 3 and a ramp function 0 if t lt t t t u t 4 C L if tr lt t lt te ta li 1 if
80. constant break points where integration is restarted with initial tolerances Constant break points are permitted only if NBPV 0 Unformatted input of NBPC rows each containing one time value in increasing order that represents a break point of the right hand side of a PDE A user has the alternative option to implement all model functions in form of Fortran code and to link his own module to the object file of PDEFIT Information about the remaining files to be linked in this case is found among the initial comments of the file PDEFIT FOR that contains the main program The model data i e fitting criterion system equations bounds and initial values must be provided by a user defined subroutine called SYSFUN SUBROUTINE SYSFUN NPAR MAXPAR NPDE MAXPDE NCPL MAXCPL NMEA MAXMEA NRES MAXRES PAR U U0 UX UXX UP V VO VP C CX FIT G DG X T IAREA LEFT RIGHT IFLAG FLUX FLUXX FLUXU FLUXUX FLUXPX The meaning of the arguments is as follows 37 NPAR MAXPAR NPDE MAXPDE NCPL MAXCPL NMEA MAXMEA NRES MAXRES PAR MAXPAR U MAXPDE U0 MAXPDE UX MAXPDE UXX MAXPDE UP MAXPDE V MAXCPL VO MAXCPL VP MAXCPL Number of parameters in array PAR Dimensioning parameter must be greater or equal to NPAR Number of functions on right hand side of partial differential equa tions Dimensioning parameter must be greater or equal to NPDE Number of coupled differential algebr
81. contain a variable or constant as a parameter If we assume that T has previously been declared as a variable a valid statement could look like FUNCTION OBJ OBJ F T The resulting approximations for piecewise constant functions piecewise linear func tions or piecewise cubic spline functions are depicted in Figures 5 1 5 3 Whereas the cubic spline approximation is twice differentiable on the whole interval the other two approximations are not differentiable at the break points and PCOMP uses the right hand sided derivatives instead Figure 5 1 Piecewise Constant Interpolation Macros PCOMP does not allow the declaration of subroutines However it is possible to define macros that is arbitrary sequences of PCOMP statements that define an auxiliary variable to be inserted into the beginning of subsequent function declaration blocks Macros are identified by a name that can be used in any right hand side of an assignment statement MACRO identifier followed by a group of PCOMP statements that assign a numerical value to the given identifier This group of statements is inserted into the source code block that con tains the macro name Macros have no arguments but they may access all variables constants or functions that have been declared up to their first usage Any values as signed to local variables within a macro are also available outside in the corresponding function block If we assume that x is a varia
82. data do not exist Thus the subsequent table contains the actual number lt l of terms taken into account in the final least squares formulation The data fitting function h p y p t c t c depends on a concentration parameter c and in addition on the solution y p t c of a system of m coupled ordinary differential equations with initial values f g amp Filpy t c qw0 mw po Ym Fmlp Y t C Yml0 Ym p C Without loss of generality we assume that as in many real life situations the initial time is zero The initial values of the differential equation system y p c yo p c may depend on one or more of the system parameters to be estimated and on the concentration parameter c The resulting parameter estimation problem can be written in the form min Dia Dita x wi Oo gp 156 have P pe R gp 0 J use 5 g p 20 j m 1 m P lt p lt Dy gt Again we have to assume that model functions h p y t c and g p are continuously differ entiable functions of p k 1 r and j 1 my and that the solution y p t c is also a smooth function of p All test problems based on ordinary differential equations are listed in Table B 4 We do not list additional information about switching points or boundary values for example 18 p nurmuoo X sr a a q cor IS vez IS 9c 1 GS IS pez a a ag a q ag Cer a IS IS vea ag 971 a uou oci 808 IS v
83. degenerate parabolic equations Archive for Rational Mechanics and Analysis Vol 96 55 80 Fu P C Barford J P 1993 Non singular optimal control for fed batch fermentation pro cesses with a differential algebraic system model Journal on Process Control Vol 3 No 4 211 218 F hrer C 1988 Differential algebraische Gleichungssysteme in mechanischen Mehrkorpersystemen Theorie mumerische Ansdtze und Anwendungen Dissertation Technical University of Munich F hrer C Leimkuhler B 1991 Numerical solution of differential algebraic equations for constrained mechanical motion Numerische Mathematik Vol 59 55 69 Fujita H 1975 Foundations of Ultracentrifugical Analysis John Wiley New York Galer A M Crout N M J Beresford N A Howard B J Mayes R W Barnett C L Eayres H Lamb C S 1993 Dynamic radiocaesium distribution in sheep measurement and mod elling Journal of Environmental Radiology Vol 20 35 48 Gallant A R 1975 Nonlinear Regression American Statistics Vol 29 No 2 73 81 Gallant A R 1987 Nonlinear Statistical Models John Wiley New York 11 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 Ganzha V G Vorozhtsov E V 1996 Numerical Solutions for Partial Differential Equa tions CRC Press Boca Raton New York London Tokyo Gear C W 1990 Differential algebraic equations indices and integral alge
84. differential equations SIAM Journal on Scientific Computing Vol 21 No 6 2295 2315 Mattheij R Molnaar J 1996 Ordinary Differential Equations in Theory and Practice John Wiley Chichester UK Mattie H Zhang L C van Strijen E Razab Sekh B Douwes Idema A E A 1997 Phar macokinetic and pharmacodynamic models of the antistaphylococcal effects of Meropenem and Cloxacillin in vitro and in experimenatal infection Antimicrobial Agents and Chemotherapy Vol 41 No 10 2083 2088 Maurer H Weigand M 1992 Numerical solution of a drug displacement problem with bounded state variables Optimal Control Applications and Methods Vol 13 43 55 Mayer U 1993 Untersuchungen zur Anwendung eines Einschritt Polynom Verfahrens zur Integration von Differentialgleichungen und DA Systemen Ph D Thesis Dept of Chemical Engineering University of Stuttgart Mayr L M Odefey C Schutkowski M Schmid F X 1996 Kinetic analysis of the unfolding and refolding of ribonuclease T1 by a stopped flow double mixing technique Biochemistry Vol 35 5550 5561 Meadows D H Meadows D L Randers J 1992 Beyond the Limits Chelsea Green Post Mills Meissner E 2000 Messung von kurzen Konzentrationsprofilen mit Hilfe der analytischen TEM EDX am Beispiel der Bestimmung von Diffusionskoeffizienten fiir Mg Fe Interdiffusion in Olivin Dissertation Faculty of Biology Chemistry and Geological Sciences University of Bayreuth Mie
85. e h p ejes q rer tk and h p q e e t are required to get the mixed second order derivatives 3 18 The perturbation tolerance e should not be chosen too small Depending on the condition number of the information matrix even large values like e 0 01 or even e 0 1 are applicable and lead to stable solution processes subject to a surprisingly small optimality criterion Example 3 4 MICGROWX Y Banga et al 22 consider a design problem based on an unstructured microbial growth model to determine feed rate profiles in fed batch bio reactors They mention that numerical instabilities prevent application of gradient based optimization procedures Instead the use a stochastic search algorithm The process is described by two differential equations and the integration of an input function C sin m Cs eu 0Cx Fin t V 3 Cs 0 Ce 5 Cx Cy 3 20 Cy uy F t Cx 0 x HUX t V x Vo where lg _ E N Yxs m Com Y W Csin CR and m 029 Csin 500 Yxs 0 47 V 7 and um 2 1 F is an input control function chosen very close to the optimal solution found be Versyck 526 Kp Ki are model parameters to be estimated and initial values C9 C are design parameters It turns out that K is very difficult to estimate Starting from some reasonable initial guesses Kp 10 K 0 1 C 40 C 10 artificial measurements are generated and perturbed by a uniform error of 5 Then confi
86. equations is treated as general nonlinear programming problem with equality con straints A minimum norm solution is computed by the sequential quadratic programming method NLPQLP of Schittkowski 427 440 449 The initial values given for the algebraic equations are used as starting values For reasons outlined in the previous section it is possible that the right hand side of an DAE becomes non continuous with respect to integration time Thus it is possible to supply an optional number time values where the integration of the DAE is restarted with initial tolerances for example with the initially given stepsize The integration in the proceeding interval is stopped at the time value given minus a relative error in the order of the machine precision Break or switching points are either constant or optimization variables to be adapted by the optimization code 3Copyright 2004 Ernst Hairer 4 6 Partial Differential Equations The underlying idea is to transform the partial differential equations into a system of ordinary differential equations by discretizing the model functions subject to the spatial variable z This approach is known as the method of lines see Schiesser 420 For the i th integration interval of the spatial variable we denote the number of dis cretization points by n i 1 ny We proceed from uniform grid points within each interval and get a discretization of the whole space interval from x to xg To approxim
87. estimated are used as break points in the code that defines the right hand side and the initial conditions In this case the number n must be inserted If n gt 0 and constant break points exist in the corresponding table then these values are ignored Copyright 2004 Ernst Hairer 29 Baba ne Number of Break Points to be PT A Optimized 1 8 Constant Break Points Figure 7 15 Break Points 30 Consistency Parameters x Output Flag for Subproblem i y Ca 4l Iteration Bound for Subproblem 50 E A Final Accuracy for Subproblem 1E 10 Ld Figure 7 16 Consistency Consistency Parameters To achieve consistency of initial values the corresponding system of algebraic equations is solved by the general purpose nonlinear programming method NLPQLP see Schittkowski 427 440 For executing NLPQLP a few parameters must be set Print flag indicates the desired information to be displayed on the screen 0 nooutput at all 1 only final summary of results 2 oneoutput line per iteration 3 detailed output for each iteration A value greater than 0 is only recommended in error cases to find out a possible reason for non successful termination Maximum Number of Iterations For computing consistent initial values NLPQLP requires the maximum number of iterations to be defined here A value of 50 is recommended Termination Accuracy To compute consistent initial values by NLPQLP one has to d
88. from ao by adapting some parameters p and the control function s t It is assumed that the control variable depends only on the time variable as in most practical applications Otherwise one could try to exchange the role of t and z Vice versa it is possible that the cost function is evaluated at a fixed spatial value z so 71 that the integration is performed over the total time horizon T Ja p s falo s t u p zm t uz p z au p v t v p t t Fa t dt 2 127 with a given time dependent function f t f may depend on solution values at a given spatial variable value 7 in an integration interval defined by the index k Thus Z belongs to the k th integration area z fo ai Org eus Lon Sch respectively As a special case 2 127 contains control problems depending only on the solution v p t of a system of ordinary differential algebraic equation of the form 2 36 with initial values Another possibility is that there is only one function to be minimized with fixed time and spatial values say Js p s falo s T u p z T u p T uk p T T v p T 2 128 Without loss of generality the fixed time value is supposed to be the final integration time T As a special case we get a time minimal optimal control problem J3 p s T D where we minimize only the final integration time In the first two situations we may discretize the integral at certain spatial or time values and get immediately a le
89. from the Fisher information matrix until one of the following conditions is satisfied 1 The smallest eigenvalue of the Fisher information matrix is smaller than a threshhold value i e the given significance level 2 The maximum parameter correlations are significantly reduced 25 3 None of the above termination reasons are met and all parameters have been elimi nated Level 1 corresponds to the first eliminated variables level 2 to the second etc The final level can be assigned to several parameters indicating a group of identifiable parameters Eigen values and eigenvectors are computed be subroutine DSPEV of the LAPACK library 7 4Copyright 1992 2007 The University of Tennessee 10 Experimental design depends on two steps First we have to setup the covariance matrix and estimate its volume in our case by the diagonal elements or confidence in tervals respectively Subsequently the nonlinear programming code NLPQLP of Schitt kowski 440 440 449 is started see also Schittkowski 427 Since the sequential quadratic programming code requires the calculation of gradients of the objective function and all constraints the procedure outlined in Section 3 3 is applied It is known that the Fisher information matrix is often only semi definite and rank deficient especially in case of ad ditional equality constraints Thus the generalized inverse is computed by the LAPACK routine DGELSS based on a singular value dec
90. gt urethane ns allophante n4 and isocyanurate ns consists of three differential and three algebraic equations of index 1 T3 ooo gt ov oa ll V ri r2 r3 n3 0 0 V ra rs 5 n4 0 Vr ns 0 0 21 ni na 2n4 3N5 Nai Nealt 3 21 To Ta d where ng denotes the solvent and b Mini 1 V Ti T2 F3 T4 n1N2 ki y2 n1N3 ka 7 El H 24 Maz N2eblt Tig Nas N6ea t d neew t gt k Kn exp Eai 1 T t 1 Trepi R kan exp Ena 1 T t a 1 Tref2 R ko ke Fresa exp Eaa 1 T t 1 Trega R keg exp dna 1 T t 1 T72 R Two input feeds are given in form of non decreasing functions feed t and feed t t 0 80 and define Niea t Nalea f eed t n3ey t Na2er feeda t neg t Nabea eed t and nga t Naseb f eedy t satisfy certain bound constraints IN IN IA IA IA IA IA Mol ratios active ingredients and the initial volume have to MV lt 10 MV x 1000 MV lt 10 Ja lt 08 gaa lt 09 Jab lt 1 V lt 0 00075 96 and are connected to the remaining parameters by analytical equations MV nai F EH Naz Na2eb M Van alea gt M Wann MNa2eb gt ga nai Mi Na2Ma nas M na Mi nao Ms gaea Narea M apen Mo m Nalea M Gaeb Naren Ma Naseb Mg Na2eb Ma Va NnaM p Na2M2 p2 nagMag pg which play
91. in advance These data are to be inserted by the user In case of different concentration values then each subset of rows belonging to one concentration has the same structure i e order and column positions of the data to be read All existing experimental data are deleted before reading new ones The data input stops as soon as the end of file is reached Measurement data can be exported to an EXCEL file with extension XLS Full path must be selected where the default file name can be changed For each set of measurement data a corresponding EXCEL column is generated In addition to time concentration experimental and weight values also model function values residuals and relative errors are exported together with column headings in the first row of the spreadsheet These data can be used for example to plot data and results After exporting data EXCEL can be launched directly 10 Import of Measurement Data x Directory IESTEMPY browse Extension DAT Existing Files C TEMPSSEG_EXP DAT CATEMPASEXP DAT Lu File Name CATEMPASEO_EXP DAT HINT To select a file from the list please click on this file name then on the field Import Data in Standard Form Import Data from Selected Columns Time Values Concentration Values DataSet Value Leni i Length Data Set Value DataSet Value Value 1 10 11 21 0 0 2 12 22 El 13 23 4 14 24 5 15 2
92. in the form lt name gt FUN for model functions in the problem directory of the actual EASY FIT VodeDesion version among many others Therefore the name must satisfy the usual DOS conventions i e must be an alpha numeric string with up to 8 characters TTT aloe J File Edit Start Report Data Delete Make Utilities r f I y G eh ri ModelDesi S d TIT oaet Design Actual Problem sot Model Data Experimental Data Type a question for help x e Version 5 00 Jan 2009 Prof Klaus Schittkowski Bayreuth Germany Model Information Project Number User Name Measurement Set Date Memo Diffusion of water through soil convection and dispersi n lt o Demo Schitkowski JExpermental 14121998 Unit for X Values t Unit for Y Values Names Unit for Z Values F van Genuchten M T Wierenga P J 1976 Mass transfer studies in sorbing porous media 1 Analytical solutions Soil Sci Soc Am Journal Vol 44 892 898 Anderson F Olsson B eds 1985 Lake Gaadsj on An acid forest lake and its catchment Ecological Bulletins Vol 37 Model Type explicit explicit model functions with concentration values system system of steady state equations EE Laplace formulation of model functions with con Laplace centration values and internal backtransformation system of ordinary differential e
93. k_y exp p3 pg exp po gt K exp pa K exp ps K3 exp pe nine parameters to be estimated and three experimental data sets obtained under different temperatures and initial concentrations T zl z 0 z O za 0 zs 0 Zell 313 15 1 7066 8 3200 0 0100 0 0000 0 0 0 0131 340 15 1 6497 8 2200 0 0104 0 0017 0 0 0 0131 373 00 1 5608 8 9546 0 0082 0 0086 0 0 0 0131 In case of estimating only one data set it is obvious that there are strong internal dependen cies between p and pz p2 and ps and ps and po This is reflected by the priority listed in Table 3 6 where y 0 1 At least one of the two corresponding priorities obtained the lowest possible value To improve the number of parameters which can be identified we add up to two additional data sets see again Table 3 6 For three different data sets seven of nine parameters are considered as identifiable within one group similar to the results obtained by Majer 315 We also observe that the parameter values get more and more stabilized and some of them for three data sets are quite far away from the parameter values for one data set 3 3 Experimental Design Mathematical models describe the dynamical behavior of a system with the goal to allow numerical estimation of model parameters a user is interested in These parameters identify 89 T 340 15 T 313 15 T 313 15 T 340 15 T 340 15 T 373 p D Jk P dy D Ji p 28 10 3 0 0 3 0 0 3
94. method for numerical parameter estimation in differential equations SIAM Journal on Scientific Statistical Computing Vol 3 28 46 Varma A Morbidelli M Wu H 1999 Parametric Sensitivity in Chemical Systems Cam bridge University Press Vasile M Jehn R 1999 Low thrust orbital transfer of a LISA spacecraft with constraints on the solar aspect angle MAS Working Paper 424 ESOC Darmstadt Vassiliadis V S Sargent R W H Pantelides C C 1994 Solution of a class of multistage dynamic optimization problems 2 Problems with path constraints Industrial Engineering and Chemical Research Vol 33 No 9 2123 2133 Versyck K J Claes J Van Impe J 1997 Practical identification of unstructured growth kinetics by application of optimal experimental design Biotechnical Progress Vol 13 524 531 Verwer J G Blom J G Furzeland R M Zegeling P A 1989 A moving grid method for one dimensional PDEs based on the method of lines in Adaptive Methods for Partial Dif ferential Equations J E Flaherty P J Paslow M S Shephard J D Vasilakis eds SIAM Philadelphia Pa 160 175 Verwer J G Blom J G Sanz Serna J M 1989 An adaptive moving grid method for one dimensional systems of partial differential equations Journal of Computational Physics Vol 82 454 486 Vlassenbeck J van Dooren R 1983 Estimation of the mechanical parameters of the human respiratory system Mathematical Biosciences Vol 69 31 55 v
95. method leads to excellent initial trajectories we would hardly get by a trial and error approach This feature explains the low number of iterations of the data fitting algorithm we usually observe in practical applications The main drawback of the shooting method is that the additional variables and nonlinear equality constraints increase the complexity of the underlying data fitting problem A typical 33 Predator Prey Model 2 2 1 8 1 6 1 4 1 2 0 8 0 6 0 4 E 0 2 Figure 2 18 Final Multiple Shooting Trajectories optimization algorithm handles nonlinear equality constraints by successive linearization requiring the solution of certain subproblems where a quadratic programming or a linear least squares objective function must be minimized subject to these linear constraints see Lindstr m 288 But from 2 58 it is obvious that the Jacobian matrix of the constraints has a very special block structure that can be exploited when solving the subproblems mentioned above If we combine the constraints 2 58 in one vector glp s1 8n 91 P 81 n p Sa with m p s1 w n 5 2 61 ls qr od Sns we get Vogilp s1i Vy pn 2 62 Van Dy s Voy p Ta gt and f Vayl p 72 f Va y p T3 V59 p 81 38n E e Vol P Tr I 2 63 34 where J represents an identity matrix
96. model functions e steady state systems e Laplace transforms of differential equations 1 e ordinary differential equations differential algebraic equations e one dimensional time dependent partial differential equations e one dimensional partial differential algebraic equations To understand at least some of the basic features of the presented algorithms to apply available software and to analyze numerical results it is necessary to combine knowledge from many different mathematical disciplines for example modelling nonlinear optimization system identification numerical solution of ordinary differential equations discretization of partial differential equations sensitivity analysis automatic differentiation Laplace transforms statistics The mathematical background of the numerical algorithms is described in Schittkowski 438 in form of a comprehensive textbook Also the outcome of numerical comparative perfor mance evaluations is found there together with a chapter about numerical pitfalls testing the validity of models and a collection of 12 real life case studies Most of the case studies possess an industrial background The general mathematical model to be investigated contains certain features to apply the numerical methods to a large set of practically relevant situations Some of the most important issues are 1 More than one fitting criterion can be defined i e more tha
97. nearest experimental concentration value In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental concentration value does not exist Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or concentration parameter The number of lines in the table must coincide with the number of constraint functions defined on the model function input file either Fortran or PCOMP 32 Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 17 Constraints 33 7 2 6 Model Data for Time Dependent Partial Differential Equa tions For the execution of the numerical analysis program PDEFIT for models based on systems of partial differential equations we need some data that cannot be retrieved from the model function file or other data Number of Differential Equations Define number of partial differential equations The num ber of PDE s must coincide with the number of model functions for the right hand side on the Fortran or PCOMP input file Number of Integration Areas Partial differential equations may be defined in different areas Their total number is to be inserted here The number must coincide with the number of areas defined in the table that defines the i
98. need to store all intermediate variables P 1 Pm By investigation of the above program for evaluating a function value f p we realize immediately that in a very straightforward way the gradient V f p can be evaluated simul taneously If we know how the derivatives of the elementary functions can be obtained the only thing we have to change is the inclusion of another program line for the gradient update by exploiting the chain rule In a natural way we denote the resulting approach as forward accumulation Definition 5 2 Let f be a differentiable function and f be a sequence of elementary functions for evaluating f with corresponding index sets Jii n 1 m Then the gradient V f p for a given p IR is determined by the following program For Lose TE let Vpi ei For t n 1 m let din k Ji Vp je i D Se Op Pj Let f p Pm Vf p VPm Here e denotes the i th axis vector in IR i 1 n Again the evaluation of gradients can be performed by suitable stack operations reducing the memory requirements The complexity of the forward accumulation algorithm is bounded by a constant times n the number of variables In other words the numerical work is the same order of magnitude as for numerical differentiation 5 2 Input Format for PCOMP The symbolic input of nonlinear functions is only possible if certain syntax rules are sat isfied The PCOMP language is a subset of Fortran with a f
99. not exist Corresponding concentration value at which a constraint is to be evaluated Note that concentration values are rounded to the near est experimental concentration value In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental concentration value does not exist function input file either Fortran or PCOMP 14 The table allows to define time and concentration Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 7 Constraints 15 7 2 2 Model Data for Steady State Equations For the execution of the numerical analysis program MODFIT that estimates parameters in dynamical systems of equations we need some integers that cannot be retrieved from the model function file or other data Systems of nonlinear equations must be solved for each function and or gradient evaluation required by the parameter estimation method moreover for each experimental time and concentration value separately The system is treated as a general mathematical optimization problem and solved by the Fortran code NLPQLP see Schittkowski 427 440 449 The starting values of an optimization cycle must be predetermined by the user in the input file for model functions They may depend on the parameters of the outer optimization problem Steady Stat
100. of batch reactors by iterative dynamic programming Journal of Process Control Vol 4 No 4 218 226 Luus R 1998 Iterative dynamic programming From curiosity to a practical optimization procedure Control and Intelligent Systems Vol 26 No 1 1 8 Luus R 2000 Iterative Dynamic Programming Chapman and Hall CRC Boca Raton London New York Washington Luus R 2002 Global optimization of Yeos optimal control problem Proceedings of the IASTED Conference Cancun Mexico 71 74 Luyben W L 1973 Process Modeling Simulation and Control for Chemical Engineers McGraw Hill New York Luyben W L 1990 Process Modeling Simulation and Control for Chemical Engineers McGraw Hill New York Machielsen K C P 1987 Numerical solution of optimal control problems with state con straints by sequential quadratic programming in function space CWI Tract Amsterdam Madsen N K Sincovec R F 1976 Software for partial differential equations in Numerical Methos for Differential Systems L Lapidus W E Schiesser eds Acedemic Press New York Mahdavi Amiri N 1981 Generally constrained nonlinear least squares and generating non linear programming test problems Algorithmic approach Dissertation The John Hopkins University Baltimore Maryland USA Majer C 1998 Parametersch tzung Versuchsplanung und Trajektorienoptimierung f r verfahrenstechnische Prozesse Fortschrittberichte VDI Reihe 3 Nr 538 VDI D sseld
101. of corresponding size Note that the evaluation of each single derivative matrix V y p Ti 1 requires differen tiation of the corresponding ODE subsystem subject to initial values since each s is the initial value for computing y p t at t 741 i 1 ns 1 However the whole system must be integrated in any case to compute the fitting criteria Gradients V y p Ti 1 are either obtained by numerical differentiation or any other approach The special block structure belongs to linear equality constraints in the subproblem so that the mn artificial shooting variables can be eliminated before starting the corresponding solver 2 5 6 Boundary Value Problems So far we discussed only initial value problems where first order differential equations with initial solution values at time t 0 are given in the form y p 0 c yo p c However there are many applications where the solution must pass also another point say at a final time T with solution value y p T c yr p c But the satisfaction of the additional boundary condition is only possible in the following situations e The underlying differential equation is a higher order system usually of second order where boundary values for the left and the right side are given e n case of a system of first order equations there are either left or right boundary values e There are additional model parameters of the system to be adapted so that a boundary value for
102. of iterations is required Usu ally a relatively small number of iterations is performed However numerical instabilities could require a larger bound 50 Maximum Number of Line Search Iterations for Solving Nonlinear Equations An internal line search is performed which requires additional function evaluations One has to define a rea sonably small bound 8 Print Flag for Solving Nonlinear Equations The output generated by the code NLPQLP consists of an iteration summary obtained for 2 3 or 4 or of an final optimization summary for 1 It is recommended to suppress all output by inserting the value 0 Termination Accuracy for Solving Nonlinear Equations It is recommended to use a rela tively small termination tolerance for NLPQLP to get very precise function and gradient values for the outer optimization algorithm 1 0E 10 If however some warnings and errors are reported by MODFIT one should try a larger value Constraints Restrictions are allowed for explicit model functions and can be formulated in form of equality and inequality constraints with respect to the parameters to be optimized and some of the given experimental time and concentration values Where the total number of constraints can be retrieved from the subsequent table the number of equality constraints must be supplied Equality restrictions must be defined first in the input file for model functions The table allows to define time and concentration values for
103. of the Fortran code in the reverse mode it is advisable to use labels between 5000 and 9999 The lt label gt part of the CONTINUE statement has to be located between columns 2 and 5 of an input line Together with an index the GOTO statement can be used for example to simulate DO loops which are forbidden in PCOMP i 1 s 0 0 6000 CONTINUE S s a i b i i iti 12 IF i LE n THEN GOTO 6000 ENDIF Whenever indices are used within arithmetic expressions it is allowed to insert polynomial expressions of indices from a given set However functions must be treated in a particular way Since the design goal is to generate short efficient Fortran codes indexed function names can be used only in exactly the same way as defined In other words if a set of functions is declared by FUNCTION f i i IN index then only an access to f i is allowed not to 1 or f j for example In other words PCOMP does not extend the indexed functions to a sequence of single expressions similar to the treatment of SUM and PROD statements In PCOMP it is allowed to pass variable values from one function block to the other However the user must be aware of a possible failure if in the calling program the evaluation of a gradient value in the first block is skipped One should be very careful when using the conditional statement IF Possible traps that prevent a correct differentiation are reported in Fischer 149 and are to be illus
104. on the right hand side V is not allowed to be altered within the subroutine Function values to be evaluated in the case of IFLAG 5 for initial values of the coupled differential algebraic equations Function values to be evaluated in the case of IFLAG 4 for right hand side of the coupled ODE S 38 C MAXPDE CX MAXPDE FIT MAXMEA G MAXRES DG MAXRES MAXPAR X T IAREA LEFT RIGHT IFLAG Array with boundary or transition values in case IFLAG 6 or IFLAG 7 The vector has to contain the computed values of the partial differential equations at an internal transition point X or at an external boundary point when leaving the subroutine Array with boundary or transition values in case of IFLAG 7 The vector must contain the computed values of the derivatives of the differential equations for the spatial variable X at an internal transition point X or at an external boundary point when leaving the subroutine Function values to be evaluated in the case of IFLAG S for the NMEA fitting conditions of the parameter estimation problem Function values to be evaluated in case of IFLAG 9 for con straints in the parameter estimation problem Gradients of constraints with respect to parameters to be esti mated dummy parameter Value of the spatial component X at an actual discretization point Time variable T Integer variable to inform the user about the actual area Logical to inform the user ab
105. optimization variables to be adapted by the optimization code In many application models we need to compute an integral with respect to the spatial variable x for example to evaluate a mass balance a ul p x t dx 54 where the integral is taken over the j th area where the PDE is defined j 1 n and where 7 1 ny The integral is evaluated by Simpson s rule and and can be retrieved from a common block or alternatively through a special construct of the PCOMP language 4 7 Partial Differential Algebraic Equations The basic idea is now to transform the partial differential into a system of differential al gebraic equations by discretizing the model functions with respect to the spatial variable x Again we denote the number of discretization points by n i 1 nr for the i th integration interval of the spatial variable We proceed from uniform grid points within each interval and get a discretization of the whole space interval from x to zn To approximate the first and second partial derivatives of u p x t with respect to the spatial variable at a given point x we may apply any difference formula as outlined in the previous section Thus we get a system of differential algebraic equations that can be solved then by any of the available integration routines Boundary conditions have to satisfy the algebraic equations Consistent initial values are computed within the code PDEFIT where the given data
106. optional concentration parameter c The inversion is performed numerically by the quadrature formula of Stehfest 190 for example Proceeding from coefficients which can be evaluated before starting the parameter estimation algorithm we compute the expression In2 hip t c 2 ath Z c 2 17 for k 1 r The vector valued fatten pm c is a numerical approximation of the inverse Laplace transform of H p s c subject to an accuracy given by the number q and defines our fitting criterion It is recommended to use q between 5 and 8 Any smaller value decreases the required accuracy any larger value introduces additional round off errors However numerical instabilities must be expected in case of highly oscillating functions A particular advantage of the above formula is that we get easily the gradient of the fitting function subject to the parameters to be estimated if derivatives of the Laplace transform V Hi p s c are available gd iln2 l V hx p t c F Lay Hyp 0 gt 2 18 M deis now from measurements t Cj V5 and weights w le and k 1 r we get the data fitting problem min XX Xia Xa wh help ti ci 95 General nonlinear constraints are omitted for simplicity Example 2 3 LKIN L3 The formulation of a data fitting model in the Laplace space is illustrated by a simple test case see also Example 2 2 A linear ordinary differential equation describes a kinetic process in the form A E ne IR
107. or by pushing the corresponding button Alternatively the Edit command allows to go directly to the corresponding area i e an input field a subtable a subform or a file to be edited EASY FIT Pesis js delivered with two editors an internal form EASY FIT and an external executable file with syntax highlighting EDITOR EXE Both allow direct pars ing of PCOMP code or compilation and link of Fortran code In case of PCOMP input the corresponding parser can be started directly by an editor command Otherwise Fortran code must be generated but a direct compilation and link is also possible The order by which variables and functions are to be inserted is predetermined by the underlying model structure Start Report Data Delete Make Utilities General problem information Experimental data Initial parameter values Model Parameters Model Functions Information Diffusion of water through soil convection and Project Number Demo Unit for SA User Name Schittkowski Unit for A Figure 8 4 Edit Command 8 3 Start Command Depending on the mathematical model type EASY FIT VodelDesi n starts one of the numer ical codes MODFIT or PDEFIT The codes perform either a simulation at a given parameter set or start an optimization cycle EASY FITYVodelDesi n generates a suitable input file with all data and tolerances required by the numerical algorithm After termination the results are read in and store
108. order information Report Chemical Engineering Lab CSIC University of Vigo Spain Balsa Canto E Banga J R Alonso A A Vassiliadis V S 2001 Dynamic optimization of chemical and biochemical processes using restricted second order information Computers and Chemical Engineering Vol 25 539 546 Baltes M Schneider R Sturm C Reuss M 1994 Optimal experimental design for para meter estimation in unstructured growth models Biotechnical Progress Vol 10 480 488 Banga J R Singh R P 1994 Optimisation of air drying of foods Journal of Food Engi neering Vol 23 189 221 Banga J R Alonso A A Singh R P 1997 Stochastic dynamic optimization of batch and semicontinuous bioprocesses Biotechnology Progress Vol 13 326 335 Banga J R Versyck K J Van Impe J F 2002 Computation of optimal identification experiments for nonlinear dynamic process models a stochastic global optimization apporoach Journal of Industrial Engineering and Chemical Research Vol 41 2425 2430 Banks H T Crowley J M Kunisch K 1983 Cubic spline approximation techniques for parameter estimation in distributed systems IEEE Transactions on Automatic Control Vol AC 28 No 7 773 786 Banks H T Kunisch K 1989 Estimation Techniques for Distributed Parameter Systems Birkhauser Boston Basel Berlin Bar M Hegger R Kantz H 1999 Fitting partial differential equations to space time dynamics Physical Reviews E Vol 59
109. oue1quiour 30 17 uorsngt q uomnnos uro301d e jo uorjeredos oURIqUIOW AYU V uro qo1d oqON OZX V e2tpo N uorsngrp pue uorj2oAuoo snoouej nulis YILM I9JSU l1 SSU jos ejep ISI uoAo uorg2oAuoo Y ur urrjxopojpeur Jo SUMI puno4bDovq oo oo oO ooo CH ooo Ch CC CC On Ch Ch Ch Ch Ch Ch Ch Ch CO CH T OT QO Ch Ch CC CH ri CO CH c On On CC CO Ch CO CO OC oo OO CO on o CV CGN CG DANN CO rd rd Cl CGN Cl r rd ri r ri Cl si Cl A r r r r amp E 0 081 0c 09 0 8I 001 001 8E IT OT OT 8I 006 0 9 9 08 9cI OT 08 OF 08 0c GC Ic GS Gl N NA CN si N TNA CH gno coco G 710 CN CV Do CO i ri MN B GO C c CH TOS DSO dH LSANO ATOVLISAO dV NON HSION ASIN Add NIIN ONG NI IN dHOS GIN SNVUL IN Tadd IN Add IN LVHH IN HAYAN NAAN XHMANOON NUNT ZIN OULNA XIN LNOUVTAON AO TOW TIN CTTIN UTIIN ANVIINAN dds WAN OZMAVUHIN VAL SSVIN XWHGO LIVIN 9 Du 55 IS gS IS gS gS gS gS gS gS gS IS gS IS gS uou uou 0S 0 0S 0S 05 GS 030p ponurjuoo oor are 9p 007 Jare 12g PEI PEI ZL Wi Le 09r eer Tosp Zig Wi ZLT TG v v v vi vi ose Leg loge fas uoryenbo spreuon1 toyem puno 3 Jo uoryemyes uoryenbo spreyory Aq erpour snood ysnory 310dsuer pmp amp reuorjejs uoN uoryenbo spreypryy toyem punos jo uorjeanj3eg uorjenbo 339194
110. p 0 1 and p 0 05 Subsequently a noise of 5 is added randomly to the data In other words we have n 2 l 18 L 3 and r 2 i e a set of l 84 measurements The minimization problem is min x 24 hj p ti e e Vis F ha p ti Cj 5 ne IR see 2 16 Starting from p 1 and p 0 1 DFNLP computes the solution pt 0 10014 ps 0 04987 after 26 iterations Final termination accuracy is set to 10 and the surface plot of both fitting criteria is shown in Figures 2 2 and 2 3 Linear Compartmental Model D 1 60 50 30 0 50 7 0 l I o 10 o Figure 2 3 Model Function Plot 2 3 Laplace Transforms In many practical applications the model is available in form of a Laplace formulation We want to proceed directly from the Laplace transform and to compute its inverse internally by a quadrature formula proposed by Stehfest 190 The advantage of a Laplace formulation is that the numerical complexity of nonlinear systems can be reduced to a lower level Linear differential equations for example can be transformed into algebraic equations and linear partial differential equations can be reduced to ordinary differential equations The simplified systems are often solvable by analytical considerations Let us assume that the model function is given in form of a Laplace transform say H p s c IR depending on the parameter vector p to be fitted the Laplace variable s and an
111. po 1 6 6 1 1 4 1 1 3 pa 8 7 5 28 4 5 0 3 Da 34 2 6 216 4 24 9 3 Ds 26 5 4 172 2 18 0 2 De 32 5 6 20 1 4 23 2 3 pr 10 6 6 9 3 4 9 3 3 pa 8 6 1 8 9 4 8 9 3 po 0 0002 2 0 04 1 0 04 1 Table 3 6 Parameter Values and Priorities for Example 3 3 the system under consideration and are to be verified by experiments However the ex perimental design often depends on parameters which must be set in advance to be able to measure certain output data of an experiment Examples are initial concentration of sub strates input feeds of a chemical reactor temperature distributions etc In addition our model may depend on universal physical parameters like gas constant absolute temperature or gravitational constant To determine the experimental design parameters in an optimal way we first have to find a suitable guess for the model parameters either from the literature or some preliminary experiments We have seen in the previous sections that the covariance matrix determines the confidence region of the model parameters see 3 3 Since we have now additional freedom to design an experiment we can use the design parameters to minimize the volume of the corresponding ellipsoid based on a suitable criterion To formalize the situation we denote again the model parameters by p R and the design parameters by q JR In case of a dynamic i e time dependent parameters for example a control function we assume that the control function is app
112. reports on numerical results are described in this chapter 8 1 File Command By selecting either a name from the pick list or by typing the name of an existing problem an available data fitting problem of the database can be loaded The pick list can be sorted with respect to name date model type or title either in ascending or descending order to facilitate the access to problem data see also the Save As command New problems can be generated by the second option of the File command All problem data including the model function file can be copied from one problem to another within the actual database EASY FIT 9 needs the name of a destination problem for copying the actual problem to another one If the desired problem exists in the database its name is either taken from the pick list or typed by hand The pick list can be sorted with respect to name date model type or title either in ascending or descending order to facilitate the access to problem data Otherwise a new problem is generated automatically if the name defined is not found in the database Note that all problem data including the model functions are copied and that existing data are overwritten Another option of the File command is to import data from text files with extensions DAT FOR and FUN respectively After defining the name of these files a new problem with an arbitrary user provided name is generated The format of the data file with extension DAT mu
113. sorenbs seo poure1jsuo punoasby20Q ooo co vo ei OO OG U GT O Ch GOO OOl OOGO Ch Ch Ch Ch Ch Ch Ch CH ooo CLG Oo ci GO Cl Gi r ei ri OO OO On Oneu CH OOO O O O OO O GL 08 TPST 0I G VG 00 IT 6I 6I 008 69 GG Ic 6I Ft OF S9 A8 A8 0c CO c c GO co CH ba uc LI M AM Lc EE E j oN M2 0 DAD lt IDSIHAA XHAVM ENHUISIA GOdHOSIA TOHYYUOSIA VIT OSIA DUdS SIA HOdVA ANVLINL ddV DIAL ANAAL XO6SNVUL O6SNVUL LAN VAL HO IDd L OLdL 9dL LGd4 L 8rd L 9rd L Sd L VOEdL 6LEd L LEdL CLEdL TLEdL OLEdL 8SEd L SG d L IWDU 12 a 030p epT fas uomnenb ioKeurqoojg urutZ 0 puno4DyonQ u 0 quu OT l G u NLOWSZ IWDU 13 10 2 Laplace Transforms Now we assume that the data fitting function is given in form of a vector valued Laplace transform A p s c IR depending on the parameter vector p to be fitted the Laplace variable s and an optional so called concentration parameter c Let function h p t c be a numerical approximation of the inverse Laplace transform of H p s c for instance computed by the formula of Stehfest 490 separately for each component Proceeding now from l liler E data t c yk and weights w le and k 1 r we get the parameter estimation problem k wt 1 4 7 1 min 2 Dia E wi hp ti c yk pe IR pi lt p lt Pu General nonlinear constraints ar
114. steps a6 4x 15 Output level for chosen optimization algorithm usage defined in documentation of optimization algorithm ex ecuted The output is directed to a file with extension HIS Only the residuals are displayed on screen a6 4x g10 4 OPTE1 Tolerance for chosen optimization algorithm METHOD 0 tolerance for rank determination METHOD 1 final termination tolerance a6 4x g10 4 OPTE2 Tolerance for chosen optimization algorithm METHOD 1 expected size of residual a6 4x 15 ODEPL1 Parameter for selection of differential equation solver ODEP1 1 dummy ODEP1 2 dummy ODEP1 3 dummy ODEP1 4 RADAUS implicit Runge Kutta order 5 ODEP1 5 dummy ODEP1 6 dummy ODEP1 11 IND DIR Runge Kutta 5 th order with internal sensitivity analysis 26 a6 4x g10 4 OPTE3 Tolerance for chosen optimization algorithm METHOD 1 internal scaling bound a6 4x 515 ODEP2 Order of ODE method or gradient evaluation respec tively 0 no derivatives 1 derivatives of right hand side supplied Number of algebraic equations in case of DAE Number of index 1 variables in case of DAE Number of index 2 variables in case of DAE Number of index 3 variables in case of DAE If NDAE NODE it is assumed that a steady state sys tem is to be solved 29 a6 4x 15 ODEP3 Approximate number of correct digits when gradients must be evaluated numerically by forward differences If set to zero a suitable tolerance is inter
115. t C FUNCTION gi gl zi 1 pl z2 p2 z3 p3 C FUNCTION g2 g2 z2 1 pi zi p4 C FUNCTION g3 g3 z3 1 p2 z1 t C FUNCTION zi 0 z1_0 p3 C FUNCTION z2 0 z2_0 p4 C FUNCTION z3_0 z3_0 t C FUNCTION h h p4 z2 C END C 6 4 Ordinary Differential Equations For defining variables we need the following rules 1 5 The first variables are identifiers for the n independent parameters to be estimated Pi Pns The subsequent s names identify the state variables of the system of ordinary differ ential equations 1 1 Ym If a concentration variable exists then an identifier name must be added next that represents c The last variable name identifies the independent time variable t for which measure ments are available Any other variables are not allowed to be declared Similarly we have rules for the sequence by which model functions are to be defined 1 5 The first m functions are the right hand sides of the system of differential equations the functions Fi p y t c Pm p y t c The subsequent m functions define the initial values which may depend on the para meters to be estimated and the concentration variable y p c y9 p c Next r fitting functions hi p y t c hy p y t c are defined depending on p y t and c where y denotes the state variable of the system of differential equations
116. t c 22 p t c Next r fitting functions hi p z t c h p z t c must be defined depending on p z t and c where z denotes the state variables The final m functions are the constraints g p for j 1 my if they are present in the model depending on the parameter vector p to be estimated Any other functions are not allowed to be declared The constants n r m and m are defined in the database of EASY FIT Mde Design In addition to variables and functions a user may insert further real or integer constants in the function input file according to the guidelines of the language Example We consider a simple example that is related to a receptor ligand binding study with one receptor and two ligands The system of equations is given in the form Zi l4d piz9 d po23 ps 0 z 1 p121 pa 0 23 1 poz t 0 State variables are z1 22 and za The parameters to be estimated are pi po pa and pa i e m 3 and n 4 t is the independent model or time variable to be replaced by experimental data The fitting criterion is h p z t pa 22 and we use the starting values z ps 29 pa and z5 t for solving the system of nonlinear equations c l C C Problem DYN EQ C l C VARIABLE pi p2 p3 p4 zi z2 z3
117. t gt to control the system at the right boundary t and tz are given and we measure the distance of y zn t from exp t cos x at 10 equidistant grid points within 0 and 1 The first possibility is to define a Neumann boundary condition at the right end point xg in the form Js xg t u t e t mz t Alternatively we have the possibility to treat the implicit boundary condition as a coupled algebraic equation u t e t Minn v t 0 with an algebraic variable v t Together with the trivial Neumann condition y rg t v t we get an equivalent formulation A third possibility is to consider the algebraic equation u t e t v u zn t 0 but now formulated for the Dirichlet boundary condition y xg t v t Since the first two options are more or less identical we perform some numerical tests with the original formulation with a Neumann boundary condition case A and the coupled algebraic equation for getting a Dirichlet boundary condition case B A five point difference formula is used for approximating first and second derivatives The discretized system of ordinary differential equations case A or differential algebraic equations case B is solved with error tolerance 1077 A few simulations are performed for t 0 3 and tz 0 7 with increasing number of lines n see Table 2 12 where the computed residual is listed There are no differences within integration accuracy under 11 or more lin
118. the role of nonlinear equality constraints Constant data are given for M 011911 p 1095 Trepi 363 16 Mz 0 07412 p2 809 Try 363 16 Ms 0 19323 py 1415 Tun 363 16 M 0 31234 p 1528 Ty 363 16 Ms 0 35733 pg 1451 R 8 314 Mg 0 07806 pg 1101 Model parameters to be estimated and for which some initial guesses are available are kept 5 10745 Ey 352 105 kan 8 1078 E 85 10 leg Tel Es 35 104 ko 17 107 do 1 08 The two input feed controls and the time dependent temperature are piecewise linear functions defined at 10 grid points between t 8 and t 80 The corresponding 30 support values are experimental design parameters together with the bounded parameters MV MV2 MV3 ga Jaca Gaeb Va and the parameters Nal Na2 Nas Dalea Tha2eb Nabea ANA Naseb Which are coupled by a set of seven equations mentioned above To sum up the whole optimization problem consists of 8 model parameters 47 design parameters 8 nonlinear equality constraints and 20 linear inequality constraints to satisfy monotonicity of the input feeds In addition there are 10 time values between 0 and 80 and four measurement functions ni ns n4 and ns and model variables are scaled to one The initial design is based on the data of Table 3 7 First we suppose that the parameters given above are the result of real data fitting run Experimental data are generated at the 10 OT time v
119. time values are rounded to the nearest experimental time value to avoid a re integration of the system In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does not exist c x value Corresponding spatial parameter value at which a constraint is to be evaluated The values are rounded to the nearest line number Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or spatial parameter The number of lines in the table must coincide with the number of constraint functions defined on the model function input file either Fortran or PCOMP 39 Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 20 Constraints 40 7 2 7 Model Data for Partial Differential Algebraic Equations The numerical integration of systems of partial differential algebraic equations is very similar to the solution of PDE s without algebraic equations Number of Differential Equations Define number of partial differential equations The num ber of PDE s must coincide with the number of model functions for the right hand side on the Fortran or PCOMP input file Number of Algebraic Equations Define number of algebraic equations The number of dif ferent
120. to q 10 but numerical instabilities prevent further significant improvements for q larger than 7 The residual is scaled by the sum of squared measurement values for each measurement set 10 2 4 Steady State Equations We consider now parameter estimation problems where the fitting function depends on a variable t called time a further independent model variable c called concentration the para meter vector p to be estimated and the solution z of a steady state system i e a system of time dependent nonlinear equations Again it is supposed that r measurement sets of the form oei de el Je lls be dei 2 20 are given with l time values le concentration values and liler corresponding measured experimental data Together with a fitting criterion function h p z t c we get a parameter estimation problem 2 1 2 2 or 2 3 by Lin wh hip z p ti amp te Cj z yi 2 21 where s runs from 1 to l l l r in any order The state variable z p t c IR is implicitly defined by the solution z of the system si p z t c 0 222 Smp zt 0 The equations are often obtained by neglecting the transient part of a differential equation so that the dynamical system is considered in the steady state The system functions are assumed to be continuously differentiable with respect to vari ables p and z Moreover we require the regularity of the system i e that the system is solvable and that the derivative
121. value see Eich Soellner and F hrer 132 for more details or Chartres and Stepleman 86 Mannshardt 317 Carver 80 Ellison 136 or Gear and Osterby 165 for alternative approaches In some situations however it is possible to avoid the internal approximation of discon tinuities by introducing artificial switching times that must be optimized together with the given parameters p The switching function q p y t can be avoided completely and the integration is safely restarted at the known switching times without crossing a discontinuity To apply the proposed strategy we need to know how to replace the switching function q p y t by suitable switching times and one should know a bit about the distribution of switching times for example their number and serial order But if we are able to collect some information a priori it is possible to simplify and stabilize the numerical integration as shown by the subsequent example 21 Example 2 10 DRY_FRI1 2 3 We consider a simple mass oscillator with dry friction between two bodies confer Eich Soellner and F hrer 132 The dynamical system is given by two second order differential equations 1 0 2 0 mii fui psign t t2 x 0 S Zo 2 46 1 0 l 1 mala f usign t 3 za 0 with my ma 1 fi sint and fo 0 p y is considered the unknown parameter to be estimated When integrating the above system for u 1 5 from t 0 to t 10 wi
122. value is missing if artificial data are needed or if plots are to be generated for state variables for which E data do not exist The subsequent table contains the actual number lt l of terms taken into account in the final least squares formulation The system of partial differential algebraic equations under consideration is uy Fip U Us Us Um t 5 Uny Fn P oss E ER 0 Parallel DEA 0 ELLO Uy Urg 02 where ug ui Ung and Ua usua ngana are the differential and algebraic state variables u ug u4 v IR denotes the state variables belonging to the cou pled system of ordinary differential and algebraic equations To simplify the notation flux functions are omitted Initial and boundary conditions may depend on the parameter vector to be estimated Since the starting time is assumed to be zero initial values have the form u p x 0 uo p z where u tg Ua is the combined vector of all differential and algebraic state variables For both end points x and rg we allow Dirichlet or Neumann boundary conditions u p x1 t ut pou t win zs u p out uzp r t p v t us p zg t Gino for 0 lt t lt T where T is the final integration time for example the last E time value t They may depend on the coupled ordinary differential and algebraic state variables We do not require the evaluation of all boundary functions Instead we omit some of them depend ing on t
123. we require that the shooting points coincide with the experimental time values tes Ti Ll i 1 9 and ns 9 l 10 Thus we get a least squares problem with 22 variables and 18 additional nonlinear equality constraints If we start now the code DFNLP from p 0 5 0 5 0 5 0 2 7 see above and ei yf i 1 9 k 1 2 we obtain the initial trajectories displayed in Figure 2 17 The algorithm stops after 11 iterations where the additional constraints are satisfied subject to a maximum deviation of 0 77 1077 see Figure 2 18 The computed optimal solution is p 0 984 0 980 1 017 1 023 It should be noted that the shooting method can be applied also to differential algebraic equations see Bock Eich and Schl der 53 Additional safeguards are necessary when restarting the integration at a shooting point to satisfy the consistency conditions The above example and especially Figure 2 17 show another important advantage of the shooting method The additional artificial variables s Sn require starting values for executing a data fitting algorithm say DFNLP If however the state variables yi p t 5 Ym p t are also fitting criteria i e if measurement values are available for all system variables then these values can be used as starting values for the shooting parameters If shooting times do not coincide with experimental times one could compute them for example by interpolation Thus the shooting
124. yi 0 3 12 Ya kum koyo y2 0 The priority analysis detects the redundant parameter r see Table 3 5 The starting value of the redundant parameter is not changed by the least squares algorithm Table 3 2 Confidence Intervals for 3 10 p Di i k 0 1126 0 0034 ko 0 0571 0 0022 D 102 4778 1 79 87 Table 3 3 Elimination of Parameters for 3 11 k Amin I vol Jo 1 0 37 10 4 0 89 3 2 0 12 1074 0 99 6 3 0 46 7 0 97 1 4 2578 3 1 0 5 Table 3 4 Confidence Intervals and Priority Levels for Overdetermined System 3 11 P Di i Jk ky 0 1328 0 0041 4 ky 0 8476 0 00065 2 ko 0 0314 0 00047 1 koo 0 0128 0 00094 4 D 51 2396 0 96 4 D 51 2396 0 96 3 Example 3 3 BATCH F1 F2 F3 FA4A A practical example is the kinetic model of a chemical reaction system in an isothermal batch reactor see Biegler Damiano and Blau 1 or Majer 315 t k2tgt8 Za k 2226 k 4219 kot2 8 la kotorg btute 0 5k 4mg Ta kQz4rg d 0 5k j4mg dn kizoxe d k imio 3 13 te kyxoxg kizy4zg k 41219 0 bk 129 0 r7 e 8 T9 219 Qt 0 zg Ka zz Kor 0 zo Ks zz K323 0 a Ky 27 Kz We have 88 Table 3 5 Priority Levels for Redundant System 3 12 p Di Jk ki 0 1126 2 ko 0 0571 5 D 102 4778 5 r 1 000 1 Qt 0 0131 T 342 15 k m 1 1 p SP Z T exp p ka explp 2 exp ps
125. 0 j 1 Me Ge mU j m 1 m s pi lt p lt Pu It must be assumed that all model functions hy p U Ux uzz u t and g p are continuously differentiable subject to p for k 1 r and j 1 m also the state variables and their spatial derivatives u p x t u p x t Ure p x t and v p t All test problems of our collection based on time dependent one dimensional partial differential equations are listed in Table B 6 Not listed are the number of integration areas switching times and structure of the boundary conditions Equality constraints do not exist in this case and m is therefore omitted 46 IS gS gS gS gS gS gS SS IS SS IS SS IS SS SS SS 030p ponurjuoo OLE 92 as as OLE as 94 Oc EEG eec 99z EEG eec Ton EEG EEG 867 867 Job 167 F I8 It foa eyep uueur Ty sorureu amp p ses jo suoryenbo 19 04 9qNJ 39078 UL Ire Jo moja uorn os uro3o1d e Jo uoryexedos ouedquiour Apuy seo1e OM suorjoorp XN 3juo1opgtp suoryenbo uorjooApe oA T suorjoo1rp XN 3uo1ogtp suorjenbo uorj2oApe oA T suorjenbo AJIATJISUOS PUL WI oo1nos reouip uou V YPM UOI309Ap V ongea repunoq 31 YL uorjenbo uonooApy ULIO 99 mos reouiuou Y Yy1M uoryenbo uomnooApyv Add noqazed q 1opao 3s1j uorgenbo uorgjooApy ULIO MOS YIM uorsngtp uorjooApe peojs reour uonrpuoo Krepunoq orporred yya uorooApy uorsngtp uorjooApe peojsun reourT Sot Ipu
126. 0 4 h p t 0 3 0 2 0 1 Figure 2 35 Function and Data Plot 67 Reactive Solute Transport u p x t 0 6 0 4 0 2 Figure 2 36 State Variable at predetermined time and spatial values and the solution of the coupled ordinary differential equation that is g p lt g p u p Tj ts ui p Tj Us uj p Tj 0 v p tk 2 122 for 7 Me 1 My Here the index i denotes the corresponding integration area that contains the spatial parameter x rounded to its nearest line and k the corresponding ex perimental time where a restriction is to be formulated Thus constraints are evaluated only at certain lines and experimental time values If they are required also at some intermediate points one has to increase the number of lines or the number of experimental data with zero weights Equation 2 122 is the discretized form of the dynamical constraints we want to define In a more general context our intention is to limit certain functions depending on the state variable for all time and or spatial variable value by 95b Ulp t us P z t EE 2 123 g p u p j t uz p j t uZ p v5 t v p t t gt 0 2 124 Or gu Ds 69 yee Pda EES 2 125 respectively for j me 1 My z xr xg and t gt 0 Of course these constraints may be mixed with the time independent parameter constraints 2 1
127. 00 CONTINUE DFIT 1 1 0 0 DFIT 1 2 0 0 DFIT 1 3 0 0 DFIT 1 4 1 0 DFIT 1 5 0 0 DFIT 1 6 1 0 DFIT 1 7 0 0 RETURN C C GRADIENTS OF CONSTRAINTS C 800 CONTINUE RETURN C C GRADIENTS OF RIGHT HAND SIDE OF EQUATIONS W R T Y C 900 CONTINUE DYP 1 1 1 0 X 1 Y 2 X 2 Y 3 DYP 1 2 Y 1 X 1 DYP 1 3 Y 1 xX 2 DYP 2 1 Y 2 xX 1 DYP 2 2 1 0 X 1 Y 1 DYP 2 3 0 0 DYP 3 1 Y 3 X 2 DYP 3 2 0 0 DYP 3 3 1 0 X 2 Y 1 RETURN END 26 0 35 0 3 ee _ data N Reet 0 25 0 2 0 15 x 0 1 0 05 0 0 1 2 3 4 5 6 et Figure 9 4 Data and Function Plot for Model DYN EQ Pri 9 2 PDEFIT Parameter Estimation in Partial Differ ential Equations Basically PDEFIT is a double precision Fortran subroutine and fully documented by initial comments Since most applications will probably execute the corresponding main program we describe only the format of the input data and the usage of the subroutine required for model evaluation More technical information can be retrieved from the initial comments of the main program for example link files parameters common blocks etc and from Dobmann and Schittkowski 117 Among the data generated by PDEFIT are result report and plot files that can be used for
128. 03 0 35 3924482 GEI IG EE E E E EE EE EE E EE S jolololojslololo ln o A lolo lololo ln o lolo llo lolololololoja GEI El EE E E E EE EE ooo on DEG E GIE E E E E EE EE EE EE Record alall 1173 ot fp of 1398 Number of plot points i e number of model Function values for plot interpolation Figure 7 5 Measurement Data 10 Scaling None Absolute Value Squared Value Plot Discretization E 3 Show Data ty Logarithmic Plot r 7 1 7 Scaling In addition to the individual weights that must be given by a user it is possible to select a global scaling strategy no additional scaling division of residuals by square root of sum of squares of all mea surement values division of each single residual by corresponding absolute measure ment value division of each single residual by corresponding squared measure ment value 7 1 8 Number of Plot Points Plots of model functions are generated from a number of function evaluations and a corre sponding interpolation linearly or by splines depending on the graphics system used The number of points where model function values are to be evaluated by a final simulation run is to be inserted The higher the number the more function evaluations are required the more lines are displayed for 3D plots in case of a PDE
129. 05 lt e lt l 58 p Po p dD 1 0 5 0 1 077 D 1 0 0 2 0 974 a 1 0 10 0 1 029 b 2 0 1 0 1 966 Table 2 13 Exact Start and Computed Solution and Dirichlet boundary values u p 0 t u p 1 t 0 2 106 for allt gt 0 are supposed First we try to obtain a smooth transition from the first to the second area for D D gt 1 and a 1 and define u p 0 5 t b u p 0 5 t 2 107 with b 1 as the only transition condition between both areas The surface plot is shown in Figure 2 30 We get a continuous transition but not a smooth one However there is only one transition condition the solution is not uniquely determined and the resulting solution is more or less arbitrarily generated If we require in addition that uz p 0 5 t uz p 0 5 t 2 108 we get the same solution as displayed in Figure 2 24 Next we construct a non continuous transition between both areas by b 2 We require that the flux is continuous i e that the spatial derivatives coincide at the transition line Then we construct a data fitting problem by computing simulated experimental data as done in the previous sections Time values are t 0 05 tg 0 1 tg 0 5 spatial values are 11 0 2 xo 0 4 zs 0 6 and z 0 8 The 36 data simulated subject to the parameter values of Table 2 13 are perturbed by an error of 5 96 Each integration area of the partial differential equation is discretized by 21 lines and
130. 1 0 z p c gt 0 Cai Z t c H Zma 0 Za D C 2 Without loss of generality we assume again that the initial time is zero The initial values of the differential equations yf p c Yh p c and of the algebraic equations 22 p c z p c may depend on the system parameters to be estimated and on the concentration parameter c Now y p t c and z p t c are solution vectors of a joint system of ma m differential and algebraic equations DAE The system is called an index 1 problem or an index 1 DAE if the algebraic equations can be solved subject to z i e if the matrix V G p u z t c 2 37 15 possesses full rank If ma 0 we get a steady state system as discussed in Section 2 3 In all other cases we obtain DAE s with a higher index see for example Hairer and Wanner 199 for a suitable definition and more details For simplicity we consider now only problems of index one although one of our standard implicit solver is able to solve also index 2 and index 3 problems Note that problems with higher index can be transformed to problems of index one by successive differentiation of the algebraic equations We have to be very careful when defining the initial values of the model since they must satisfy the consistency equation Gi p y p c 2 p c t c 0 Gm pu lp c 2 p c t c 0 2 38 Otherwise we have to check whether the consistency condition is satisfied before s
131. 1 ueurmngr oouooso1oudsoud uorjoeor erwy umpnpued uwa uorjep nuriog 103dri sop poqoofoid um npuoed ure q K euxogut pognduroo son ea perur quojsrsuoo umpnpuod ure q uomnsemuroj e x pur umpnpued ure q uorje nurlog c xopur umpnpued ure q d uorje nurioj T xopurt umnpued ure suoryenba ore1qo3 e OM YILM uorjerutriog e xopur unpnpued ue q USISOP eguouit 190dX9 pue jos eyep ouo o durexo oruropeoe s1ogourered jo uorjeogrjuop m pue jos jep ouo o durexo orumpeoe siogoueled jo uorjeogmuop jos 19jourered poonpoz o durexo oruropeoe srojourered jo uoryeogrquopm o durexo oruropeoe s1ojoureded Jo uorjeogrjuopm SOJITM 1ourour odeys Jo wiog eurrgd moq ut o qreur JO quouroAO A sjutod x a1q OM YIM opour quourjreduroo reou o duutg ose otdo30st OY ur s103onpuooruros 10 opour orureu amp poap amp u re odru yoyoel Sutr ooo pue suorjoeor OTULIOYJOXS SUOIIS YAM 1099891 Ye ourn oA juejsuoo u3IA ozuoq Jo uoryex1odea q Soouej3sqns 99119 10 uurn oo uorjer stuq Soouejsqns OM 10 uurn oo uornjer suq dA 1op1o pug uorjoeor pue uorsngrp opreed uornjos 3oexo g XOPUT YPM suorjenbo oreiq 3 e erjuo19pgrp e jo uro3s4g c x pur yya suoryenbo ore1qoS e erguorogtp 99 173 jo toys Ag punoasby20q N ri r ri DO DO r Cl r CY CY OO ei OO ed r r ri r G I I Du OO Y Y So TM 00 00 O CH MAON CN cO co CN CN CY 901 T G G Pus 09 99 VOG OV F 08 08 08
132. 130 MP for the Microsoft Office Access 2007 runtime version The program runs under Windows XP or higher EASY FIT Pes comes with the royalty free runtime version of Microsoft Office Access 2007 English All model functions are defined in the PCOMP modelling language to be interpreted and evaluated during run time Derivatives as far as needed are computed by automatic differentiation The full version of EASY FIT Pesis allows also the most flexible input of the underlying model functions in form of Fortran code and has interfaces for Compaq Visual Fortran Watcom F77 386 Salford FTN77 Lahey F77L EM 32 Absoft Pro Fortran Microsoft Fortran PowerStation and Intel Visual Fortran for Windows 32 and Windows 64 environments where the compiler and linker options can be altered and adapted interactively 0 1 2 Packing List Basically EASY FIT dc Pes s consists of a user interface in form of a database imple mented in Microsoft Office Access 2007 and some numerical routines The following essential files and directories are submitted Numerical codes MODFIT EXE MODFIT FOR MODFUN E FOR MODFUN O FOR MODFUN A FOR MODFUN_S FOR MODFIT INC PDEFIT EXE PDEFIT FOR PDEFUN FOR PDEFIT INC COMPILE BAT LINKER BAT Solving parameter estimation problems in explicit mod els time dependent algebraic equations ordinary dif ferential equations differential algebraic systems and Laplace transforms
133. 1997 Finite Difference Schemes and Partial Differential Equations Chap man and Hall New York Swameye L M ller T G Timmer J Sandra O Klingm ller U 2002 Identification of nucleocytoplasmic cycling as a remote censor in cellular signaling by database modeling Pro ceedings of the National Society of Sciences of the USA Vol 100 1028 1033 Sweby P K 1984 High resolution schemes using flux limiters for hyperbolic conservation laws SIAM Journal on Numerical Analysis Vol 21 No 5 995 1011 Tanaka Y Fukushima M Ibaraki T 1988 A comparative study of several semi infinite nonlinear programming algorithms European Journal of Operations Research Vol 36 92 100 Teo K L Wong K H 1992 Nonlinearly constrained optimal control of nonlinear dynamic systems Journal of the Australian Mathematical Society Ser B Vol 33 507 530 33 503 504 505 506 507 508 509 510 511 512 513 514 515 516 Thomas J W 1995 Numerical Partial Differential Equations Texts in Applied Mathemat ics Vol 22 Springer Berlin Thomaseth K Cobelli C 1999 Generalized sensitivity functions in physiological system identification Annals of Biomedical Engineering Vol 27 607 616 Thomopoulus S C A Papadakis I N M 1991 A single shot method for optimal step compu tation in gradient algorithms Proceedings of the 1991 American Control Conference Boston MA American Cont
134. 2 Y 1 DYP 1 3 K1 C Y 1 DYP 1 4 K1 C Y 1 K1 R Y 1 K2 RETURN GRADIENTS OF INITIAL VALUES OF ODE W R T X 600 CONTINUE DY0 1 2 0 0 DY0 1 3 0 0 RETURN GRADIENTS OF FITTING CRITERIA W R T X 700 CONTINUE DFIT 1 1 DFIT 1 2 DFIT 1 3 DFIT 1 4 RETURN I OO CH O OO CH GRADIENTS OF CONSTRAINTS 800 CONTINUE RETURN 16 26 data h r t 22 20 18 16 p 14 12 J 10 E Figure 9 1 Final Trajectory for Problem DFE1 C C GRADIENTS OF RIGHT HAND SIDE OF ODE W R T Y C 900 CONTINUE DYP 1 1 Ki1 C Y 1 Ki R Y 1 K2 RETURN END Lf data Ad 00 50 0080 100 Figure 9 2 Final Trajectory for Problem SE 18 Example 9 3 VDPOL The input of data and model functions in case of a differential algebraic system is to be outlined by an example the van der Pol s equation see also Section 5 5 The model function is g z y a l y z 9 3 We choose the consistent initial values y a 2 b a 1 b and consider a and b as parameters to be estimated The fitting criteria are the solutions y and z The data input file MODFIT DAT has the following structure C EASYFIT PROBLEMS VDPOL VDPOL
135. 20 Note that the 68 formulation of dynamical equality constraints does not make sense since they should be treated as algebraic equations Example 2 27 HEAT_R We consider again our standard test problem Example 2 17 the heat equation Ut Diss gt with diffusion coefficient D The spatial variable x varies from 0 to 1 and the time variable is non negative i e t gt 0 Initial heat distribution for t 0 is u p 1 0 asin rz for all x 0 1 Dirichlet boundary values are the same as before confer 2 80 and the parameter vector p D a is to be estimated subject to the same simulated experimental data computed for Example 2 17 with exact solution p 1 1 and 21 discretization lines Now we consider the second spatial derivatives u p x t see Figure 2 37 The biggest curvature is observed at the point t 0 and x 0 5 with u p x t 10 Our goal is to prevent the curvature from achieving any value below 8 Thus we formulate one dynamical constraint g p Ure p 2 0 8 gt 0 or after a suitable discretization g p asl p Tj 0 8 gt 0 for j 1 9 with x 0 17 Note that the constraints are violated at the exact solution p 1 1 7 for which experimental data are simulated Starting from po 2 2 DFNLP computes the solution p 0 92 0 81 7 in 10 iterations with termination accuracy 1078 The fifth constraint becomes active i e gs p 0 17 10 5 and the remaining ones a
136. 271 290 Schittkowski K 1980 Nonlinear Programming Codes Lecture Notes in Economics and Mathematical Systems Vol 183 Springer Berlin Schittkowski K 1983 On the convergence of a sequential quadratic programming method with an augmented Lagrangian search direction Mathematische Operationsforschung und Statistik Ser Optimization Vol 14 197 216 Schittkowski K 1985 86 NLPQL A Fortran subroutine solving constrained nonlinear pro gramming problems Annals of Operations Research Vol 5 485 500 Schittkowski K 1987 More Test Examples for Nonlinear Programming Lecture Notes in Economics and Mathematical Systems Vol 182 Springer Berlin 28 429 430 431 432 433 434 435 436 437 438 439 440 441 442 Schittkowski K 1988 Solving nonlinear least squares problems by a general purpose SQP method in Trends in Mathematical Optimization K H Hoffmann J B Hiriart Urruty C Lemarechal J Zowe eds International Series of Numerical Mathematics Vol 84 Birkhauser Boston Basel Berlin 295 309 Schittkowski K 1992 Solving nonlinear programming problems with very many constraints Optimization Vol 25 179 196 Schittkowski K 1993 DFDISC A direct search Fortran subroutine for nonlinear program ming User s Guide Dept of Mathematics University of Bayreuth Germany Schittkowski K 1994 Parameter estimation in systems of nonlinear equations Num
137. 30sp 7 Sap 19 10 ejep suorjoeor 988 pue MOTS 1039891 q jeq eurroq30sq O Sop Op 10 e3ep suorjoeor 3sej pue MOTS 1039891 938 eurroq3osq SJOS VILVP OM suorjoeor jsej PUL AO S 10J9891 UIQ eurioq30sq Suorjoeoi jsej pue MOTS 103 01 q3jeq peurroq30sp e3ep poeje nurrs sooraop oso roe oseuqd o ur uorjed3uoouoo ojex3sqng S SIA D OSOIOR oseqd o ur uorje1quoouoo ojex3sqng Sjurerijsuoo q3r4 3oqoz reuerjd ut 0A T punosbyanq M5 CN CN MN si ri ri si hi ai hi bh bh ri ri OWN WOH O c n n N CN CO o o cO cO cO cO cO cO dO dO cO CN GO bh GO c Qo co bh tT lt H N OV OV 001 GI LLG VII UI 00 GL VG 99 84 US 8c VOG v8 0cI VoL v8 TET VOG 8cI 9 6c 08 E CO iO B CO dO iO Gi Gi CN CN CH vd ri ho pv On 00 Om CO 00 a Oo CT avd cl avd IT Ava XM dvd DO SNAUNOD S IHO ds LYO Od Tddnd ONOd VAYVHOLVE cX HO LLVH IX HOLVH YA HOLVdH Cd HOLVd 61 HOLVd LT HOLVd S4 HOLVA cH HOLVA Td HOLVd 4 HO LVd HOLVd SIOSOYAV TOSOUAV HOH ONIG IWDU suoryenbg 9re1q93 y Pup T 21901 42 penurquoo gS da GN gS SS IS IS IS IS IS IS IS dd SS SS SS TS SS SS SS SS SS TS TS TS x SS 020p ere ere cre ere IT aze 9T 61 IT IT IT fas SoAIso dxo gurjeuojop ur ouoz OTRO USISOP equourtjodxo YIM 30qo1 ereog uorjung OIJWOD UAB pue s1ojourered MOJ YPM 30qo1 LILI ulo3s As Ki1ojeurdso
138. 5 6 16 26 F 17 27 8 18 28 3 13 29 10 0 D 20 30 0 0 Figure 8 8 Import of Experimental Data Experimental data can directly be imported from an EXCEL spreadsheet i e a file with extension XLS For each set of measurement data i e time concentration measurements and weights a cell range must be defined The range should not cover more than one spreadsheet column The order of concentration data must be the same as required by EASY FIT ModelDesign Time values are required in any case and all existing experimental data are deleted before reading new ones Note that EXCEL Spreadsheet Version 97 is supported IN D Export of Experimental Data to EXCEL MG te C ATEMPSSOILXLS Figure 8 9 Export of Experimental Data to EXCEL 12 Import of EXCEL Data E xi Directory IL TEMPI browse Existing Files File Name HINT To select a file from the list please click on this file name then on the field start end import Data Time Values az A767 As Concentration Values E2 E767 Values Weights Values Weights Values Weights DataSet start end start end Data Set start end start end Data Set start end start end B2 B767 S 11 E 21 i B 12 E S 22 E 13 i 23 E 14 E E 24 E E 15 z 25 y E 16 26 E 17 S 27 3 18 z 28 S E 19 t 29 E 20 30 Figure 8 10 Import of Experimental Data from EXCE
139. 5 van der Pol s equation electrical circuit Demo Schittkowski Simulation t NPAR NRES NEQU NODE NCONC NTIME NMEAS NPLOT 50 NOUT 0 METHOD 1 OPTP1 40 OPTP2 8 OPTP3 2 OPTE1 1 0E 07 OPTE2 1 0E 01 OPTE3 1 0E 02 ODEP1 ODEP2 ODEP3 ODEP4 ODEE1 1 0E 09 ODEE2 1 0E 06 ODEE3 1 0E 04 a 5 0E 01 1 001 1 b 1 5 2 001 SCALE sl H Oo OH OO H O O Oc ano oo 19 0 0 2 000 1 0 6 667E 1 1 0 2 0E 1 1 858 1 0 7 575E 1 1 0 4 0E 1 1 693 1 0 9 069E 1 1 0 6 0E 1 1 485 1 0 1 233 1 0 8 0E 1 1 084 1 0 6 200 1 0 NLPIP 0 NLPMI 50 NLPAC 1 0E 10 NBPC 0 The code DFNLP computed a solution in 4 iterations The optimal fit is shown in Figure 9 3 The corresponding Fortran subroutine SYSFUN needs gradients only for IFLAG J9 i e for gradients of the right hand side of the DAE subject to the state variables y and z since internal numerical differentiation is not allowed SUBROUTINE SYSFUN NP MAXP NO MAXO NF MAXF NR MAXR X Y T C YP YO FIT G DYP DYO DFIT DG IFLAG IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X MAXP Y MAXO YO MAXO YP MAXO G MAXR FIT MAXF DYO MAXO MAXP DYP MAXO MAXP MAXO DG MAXR MAXP DFIT MAXF MAXP MAXO C C BRANCH W R T IFLAG C IF IFLAG EQ 0 RETURN A X 1 B X 2 GOTO 100 200 300 400 500 600 700 800 900 IFLAG C C RIGHT HAND SIDE OF ODE C 100 CONTINUE YP 1 Y 2 YP 2 A 1 0 Y 1 2 Y 2 Y 1 RETURN C C INITIAL VALUES FOR ODE
140. 50 151 152 153 154 155 156 157 158 159 160 161 162 Findeisen R Allg wer F 2000 A nonlinear model predictive control scheme for the sta bilization of setpoint families Journal A Benelux Quarterly Journal on Automatic Control Vol 41 187 192 Fischer H 1991 Special problems in automatic differentiation in Automatic Differentia tion of Algorithms Theory Implementation and Application A Griewank G Corliss eds SIAM Philadelphia Fischer P 1996 Modellierung und Simulation der Ammonium und Nitrat Dynamik in strukturierten Waldb den under besonderer Ber cksichtigung eines dynamischen hierarchis chen Wurzelsystems Diploma Thesis Dept of Mathematics University of Bayreuth Ger many Flaherty J E Moore P K 1995 Integrated space time adaptive hp refinement methods for parabolic methods Applied Numerical Mathematics Vol 16 317 341 Fogler H S 1974 Elements of Chemical Kinetics and Reactor Calculations Prentice Hall Englewood Cliffs NJ Fraley C 1988 Software performance on nonlinear least squares problems Technical Report SOL 88 17 Dept of Operations Research Stanford University Stanford CA 94305 4022 USA Frias J M Oliveira J C Schittkowski K 2001 Modelling of maltodextrin DE12 drying process in a convection oven Applied Mathematical Modelling Vol 24 449 462 Friedman A McLead B 1986 Blow up of solutions of nonlinear
141. 6 6f VI 0 0G 08 9 TT 0cI OT 0 O oO Oci c0 Cl GO lt H Cl ei ri ri 00 c 00 n e lq co cq Ao CH SNVUL dO UATAUVO YIrIvo LSund X Yong q udound Tuadounad Ju Wound a HJ duouna ATadad Old gana asa THSsnud acssnud ANIYE NIVUd dA MOTE NUS G Id WTHAOId ALISUNIA SATLAHA SSNOTIVd AHAIA XV NUVA VUN d NIHG IV AOTIV IV 9L Du 48 m IS SOS IS SS SS gS gS gS gS gS IS gS gS IS SS SS GS 030p penurjuoo e IS c6 veg 16 Lob GIG DEL TGP VGC 16 Oct EEP 61 61 16 LOTMA 62g IcG fas sjos ejep e seore g yya oueiquiour SISATeIP YSno1y uorsngrp ojerjsqug svore OM YPM ueiqui ur SIS TeTp YSno1yY uorsnjrp oyeaysqns oue1quiour sisA erp Y3NOIYY UOISN PIP ojezjsqug juourLIodxo w Suo juorogpooo uorsngrp enu uodx yym oueiquiour ssq Ted snoouoSouroqur Hd d tr oqied q 19p O 9S11J suonjruel JIM seore OMY ur opour UNS euriopsueig 1035981 Ie nqn3 e ur ouo1A3s 0 ouozuoq 4u3o Jo uorjezruoSodp qo q oqnj Y ur uorjooAuoo uorsngit T oo1nos Je Suryeururop YIM uro qo1d onyea amp repunoq JAY esaou ojo duroo pue uoppns yeoiq wep pozi eopm Grp uu urorq YM Me UOTYRAIOSUOD AQNO uorjenbo osem uro qo1d o1juoo peurd Tes o qeouriod qyym o1njxrur pmbr amp reurq JO SMop 3uor1n 103unoo qtss duroour OM T UOPTOS ouo yya uorjenbo 19GUIPIOITIS DINO erpata sn
142. 8 implicit Runge Kutta method of order 5 called RADAUS see Hairer and Wanner 199 Some parameters are to be set that cannot be retrieved from the model function file or other available data Final Absolute Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the absolute global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible It is recommended to start with a relatively large accuracy e g 1 0E 6 together with a low number of iterations and to increase the accuracy when approaching a solution by restarts Final Relative Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the relative global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible Again it is recommended to start with a relatively large accuracy for instance 1 0E 6 and a low number of iterations and to increase the accuracy when approaching a solution by restarts Initial Stepsize for Solving Differential Equations Define initial stepsize for differential equa tion method used The parameter is adapted rapidly by internal steplength calculation Bandwidth of Jacobian of Right Hand Side Implicit methods require the evaluation of the Jacobian of the right hand
143. 8 Numerical Methods in Multibody Dynamics Teubner Stuttgart Eigenberger G Butt J B 1976 A modified Crank Nocolson technique with non equidistant space steps Chemical Engineering Sciences Vol 31 681 691 134 135 136 137 138 139 140 141 142 143 144 145 146 147 Ekeland K Owren B Oines E 1998 Stiffness detection and estimation of dominant spectra with explicit Runge Kutta methods ACM Transaction on Mathematical Software Vol 24 No 4 368 382 Elezgaray J Arneodo A 1992 Crisis induced intermittent bursting in reaction diffusion chemical systems Physical Reviews Letters Vol 68 714 717 Ellison D 1981 Efficient automatic integration of ODEs with discontinuities Mathematics of Computational Simulations Vol 23 12 20 Elnagar G N Kazemi M A 1998 Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems Computational Optimization and Applications Vol 11 No 2 195 213 Endrenyi L ed 1981 Kinetic Data Analysis Plenum Press New York Engleborghs K Lust K Roose D 1999 Direct computation of periodic doubling bifurca tion points of large scale systems of ODE s using a Newton Picard method IMA Journal of Numerical Analysis Vol 19 525 547 Engquist B 1986 Computation of oscillatory solutions to partial differential equations in C Carasso P A Raviart D Serre eds Nonlinear Hyperbolic Problem
144. 9 A9 _FME oy Aq UOTYLTNUIIS ILTOATIOSOY uoryenba uorsngtp uorjoeoqq uoryenba UOIsnyIp uolyoevey sum u nb re ur uou o3e10uoSo q Surqouonb reouip uou ojexouoSo q so1od Zo Tod rom otemt pmbr jo ood e woy 1odea Jo uoryerodeay uo1A3s pue Jje Ad4owuq urX u9 ur JO uorjezrrour amp qpodoo epey ssoooid uorjeziiourtAjod Jo u3suo ureuo 9194 dsoye13s oy ur uorjnjod LSS suorjrpuoo amp repunoq orporrod u3r uorjenbo or oquod q or oqu req Soso1so1d e3uop ur soorrjeur gurpersop x nq WO oseopo1 pojerpour uorsngt T uro qoud o1juoo eurrjdo orpoqereq uro qoud o1juoo eurrjdo orpoqeeq 3599 Ayriqegnuopr uorjenbo oroqereq 3599 Ayriqegnuopr uorjenbo oroqereq 3599 Ayrqegnuopr uorjenbo oroqereq 3599 Ayriqegnuopr uorenbo oroqereq 3599 Ayriqegnuopr uorenbo ogoqereq 3891 At map uonenbo oroqereq ULI9 SNUIS snoouoSouiroqut YIM Ad opoqered uro qo id o zjuoo peurrdo or oqereq Sjuouoduioo oA Jo uorydiosop uorjdiospe yym poq paxoed e tom pm 4 puno4bDovq T O T Ch O Q Ch Ch Ch Ch CC CH CC OC OT OC Ch Ch Ch Ch Ch CH ri OO r OO ei ri CO OO Ch CO OO Ch si CC CC OC Ch Ch Ch Ch Ch CH NAN ei ri r r r c rd rd SH AN m rd rd Cl r ri r nm rd c 9 207 0cI OT GET S 9 TS 8 Ch qI 8 0 GI 8y 86 LG OT 0 0 09 v8 OT F N 00 MODO Go P lt Gil r ri Cl c hi oae G IT IG NAX HOTA Ld HOIM NOW HOM IOAYASHY CAT Va LAIT VAN ZHONANO THON
145. 9rcd L Vvcd L Gr cd L Ir GIcd L S06d L 9 Du 11 ponurjuoo gS gS a a a a IN a SS a a a SS a 9uou d ouou d ouou ouou ouou 9uou 9uou 9uou Oo N CO uou pop 8z1 pez SIL 8c 666 666 GGG GGG 666 GGG 8c 8c 8c 8c 8c la Ela Glas fas uonnqrnstrp MEITA uorjyenbo A A Jo uornjos 3rot dx d sjos ejep Up uoisso1891 jISOOSTA Z OS jep uorsso1So1 AJISOSSIA T 998 eyep uorsso1do1 JISO9SIA SOOULISGNS DIISB 9 O9STA JO uorjoung Kiouro N e1300dS DIJSP 9OISIA umriqimbo pmbr 1odeA ayeys Apvojs s1039891 re nqn3 pue ue jo uostreduroy S92 03 erxe Surjnduroo 103 uorjeurrxoddde orrourouoStg OAID2 DOIT eyep egueurriedxo Sap 06 Ye 1033rurq Sop 06 1 1097 GO BOOT YPM uomnounj 3s9T ULIOJ ut suoryenbo 99 193 JO OLMOS eqo 5 uro qo1d umrrqi mbo peorumoq Uorje nuto S33oqp Uorpuny eueueq s ADOIQUISOYH 10 erguouodxo poure1jsuo uro qoJd 9599 oruropeoe pourerjsuoo Ayrrenby uro qoJd 9599 oruropeoe pourerjsuoo Ayrrenby UIo qO Id 9599 p nzng uoso jure1jsuoo jr enbo ouo yya uro qo d sorenbs seo sw erquouodxo moz ou1oqs jo uro qo 1d 3S9 sjure1jsuoo Ajrpenbo xis uro qo id s renbs jseorq sjurezjsuoo Ayryenbout oA 9A3 uro qod sorenbs seo so qerreA ouru uro qo id sorenbs 3seo xo duro so qerreA xis uro qo1d sorenbs 4seo xo duio uorjung 9899 Surj3j e3ep erguouodxzq Suor n os 890 pue sw oreipenb moz uro qoad
146. A The control variable s is a bang bang functions jumping from 1 to 1 and vice versa at some switching points s ip Sp 76 i D Pi 1 0 2 0 2914 2 0 4 0 2914 3 0 6 0 7268 4 0 8 0 7268 5 0 9 0 8483 Table 2 18 Initial and Final Switching Points To be able to apply our data fitting software under consideration we normalize the time variable to get a state equation defined for 0 lt t lt 1 The differential equation in 2 136 becomes ty Tes 2 137 Then the least squares objective function to be minimized is 9 2 h T s t YU ulT s 1 f zi 2 VaT 2 138 1 Moreover we have to take into account additional linear constraints for the switching points When starting NLPQLP from the starting values of Table 2 18 with termination tolerance 1078 21 discretization lines and an implicit ODE solver with final accuracy 1079 we get the results of Table 2 18 after 8 iterations Obviously some of the switching points coincide in agreement with the results of Schittkowski 424 Final integration time is T 1 329 The maximum deviation of u p x T from f z at a grid point is 0 0113 State and control variables are displayed in Figure 2 44 and 2 45 respectively TT Heat Diffusion Heat Diffusion Figure 2 44 State Variable 0 8 0 6 0 4 0 2 Figure 2 45 Control Variable 78 Chapter 3 Statistical Analysis and E
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148. A108Sqo su p uoryenbo nepuery SinqzZuly c uorsuouirpoo YIM uoryeoangteq uoryenbo nepuey Sinqzury g uorsuourpoo YALA uorjeoangreq sureoq poxurT ureoq p Amo uorje nuriog quopuodop ooeds 1039891 ouo 43o00 re nqnT uorjemuriog quopuedop ourr 1039891 ouo 3oo re nqng SXstp uorjo1 2e uerio dow UL soA1no umrqrrnb euro Y SXstp uorjo1ooe ueLio dow ur soA1no urnriqipmbo euro p uro qo1d JAMAL snooueSouroqur 1opio puooog uro qod 3o qori T 19pio puooog puno b390q ooo CH ooo oo On Ch Ch Ch CH CH oo Ch CH CO OC Ch Ch Ch CO CC CH CH o E rd rd 0C ri ri ed PDH OO ri Cl Cl OO O 3 CH r ed CO Co ri GO O O Gi CU CH CH ado 3 E 8y 0c OV OV OV 89 08 0 I OT 00 08I 06 66 0c 08I GIT 0 0c 0c CO ri vi N 00 ch gie Go r ri Cl Gi oO Go Go CH MAVA MA V NAATA dahl VTA dsno MOTALO AXO OD CH ENO NDISHYHO TIIdVO ATUL dAd GI LI TOUN ala cNVAd LN Vad Z TALHOV L IALHOV A LION V LION CUIT ANG THIG NC 9t Du suoryenby ore1qoSpy PUHA eed L A NWI 62 a gS IS SS SS gS X X a IS IS IS 9uou gS 0S X gS gS uou a GS GS GS d 030p ose LLT 6 66 00 LOE 991 GTZ GTZ GTZ 6 CGV CGV AAT ots ots ureoq Y JO SUOTJVIQIA oSIOASURI JO o13u00 ATepunog ureoq peireAo rjuvo ULIOFIUN utt T 1013998 Sur ooo pue
149. AE The final m functions are the constraints g p j 1 My if they exist They may depend on the parameter vector p to be estimated Any other functions are not allowed to be declared The constants n ma Ma r and m are defined in the database of EASY FIT Model Design and must coincide with the corresponding numbers of variables and functions respectively The last n fitting variables are considered as switching points if they have been declared a priori to describe certain model changes In addition to variables and functions a user may insert further real or integer constants in the function input file according to the guidelines of the language PCOMP Example We consider a modification of van der Pol s equation given in the form y z y a y We choose the consistent initial values y b 2 b a 1 b and consider a and b as parameters to be estimated The fitting criteria are the solutions y t and z t The model input file has the following structure s susa C C Problem VDPOL C C M M C VARIABLE a b y Z C FUNCTION y t y t z C FUNCTION alg equ alg equ y a 1 y 2 z C FUNCTION yO yO b C FUNCTION zO z0 b a 1 b b C FUNCTION hi hi y C FUNCTION h2 h2 z C END C 6 6 Time Dependent Partial Differential Equations For defining variables we need the following rules
150. AMIV LAITHIV NIMM ILVdaVIdV ULINLOV AJLIALLOV ULSOOLSG ALVY ANG quo UNT HOW ANTE ULSOG AdO IWDU suoryenb ferguo1ogr Areulplg rd NWL 19 ponurjuoo gS gS 9uou E gS gS ANA 3 S la DARA a a ma ma ma ma a m t Y vez vez 90 80S 80S 897 GCE 621 06 06 pez TI II srg fas oouonbos uorjoeai xo duroo YPM 1035921 Pore 1032891 q2jeq V ur SIJU sso uorsuourt T 1032891 2jeq reoui uOU Jo O1jUO SOTA eurur o qerreA qjr uorjenbo Lm3 OA ex30 UOILLA eortr uo uorjenbo LIION ex30 UOLIA LNW qeq 3ununog sorueu p uorje ndod et19308g morreu ouoq ur so3 amp ooqdur amp T g Sjuourj Teduioo omg 19x50 q 39q JO 013u0 7 sjuourj reduroo om 19X20 q e3oq JO 1017000 sjuourj Teduroo omg 19X50 q 39q JO 013u0 7 uors1odsrp erxe Un uurn oo uorjoerjxo erguod9gpr T uro qo id enpe Arepunoq orgjogdui sy sarytie n3uls uns punozre 13 189 jo uorgour reue q odo s pue w eryuouodxo YIM SOTJOUTA uorjemossyv SOTJOULY uorjemossyv SOTJOULY uorjemossyv som ump uorjemossyv ULIO erguouodxo UJIM sonour uorerossy Sorjoun uorjemossv S9A MOI UOIjeI2OSS V S9A MOI UOIjeI2OSS V S9A MOI UOIBIDOSS V S9A MOI UOIBIDOSS V S9A MOI UOIJBIDOSS Y S9A MOI UOIBIDOSS Y S9A MOI UOIBIDOSS Y Sulyyy SAM punosbyonq oo D CC Ch CC Ch Ch CC CC Ch Ch O OT OOO O Ch Ch vn 0 CH oo T CC Ch CC Ch Ch CC 00000 O OOO O coco vn Ch CH N
151. ANO quod 100d DHHINATOd NAG ATOd NIA TIOd Odd dAHd dHOOTYdd CTULO dd DILL dd 9dVUVd sdvuvd vavuvd e dvuvd cqvuvd IdVHVd NIS HVd TALO HVd aag MOVd 9 Du 56 dd IS IS gS gS gS IS gS gS IS gS IS IS 030p ponurjuoo F F PO FS V i F eer Trer 6 EEP EEP EEP EEP EEF EEP CGY CGY T9 FII LLG 691 687 T67 T67 866 16 T67 T66 OLE fas uSIsop Term ye S RAIOYUL oouopguoo ayen eaH p doLLS uorngrp eurropsued Stop feguourtrodx p amp GALS uorsngrp eurropsued S1oyoure1 ed jweogruSrs YIM Sur VPA 1 JALS uorngrp eurropsuez sdojourered YIM err 38114 0 ALS uorsngrp peuriopsuei uorsngtp peuriopsueim uorsngrp peuriopsueiT uorsngtp YALA uorjoeoz doj3s o gutg uorjn os uorsroo qung urq exo uorjenbo uopor outg uonn os uojrpos ury exo uorjenbo uop1or outg uorneuLno pueq reoug UMOUY uornjos 32exo yuox dreys YIM Add ouoz SUIT JOS UTYILM sorureu p pr og os nd osdi o tutes Jo uorgjooApy oueIquiour YSNOIY uorngos oues Y ur Snap Jo uorsngt T arsojdx pijos JO poy ses ordorjA od e Jo suoryenba Ipmy 10 uro qod uueurong S POS poys Zut OAD OUIS IOYSO NYS YALA suoryenba Irmy 10 uro qo1id Gar suoryenbo Iom 10 uorjo e1ojur 9oA A Kdoijuo xoous yya uro qoid uueumm suor enbo Zomm 103 ut qo d Omar AS1 u eUJOJUT pue Aq3tsu p mo Xer Jo uornemuoz suorgenbo Iom 103 uro qo1d u
152. B 2004 Numerical expe rience in the solution of several kind estimation problems in dynamical systems International Conference on Modeling and Optimization MODOPT 2003 Tmuco Chile 16 22 1 2004 Seydel R 1988 From Equilibrium to Chaos Practical Bifurcation and Stability Analysis Elsevier Amsterdam Shacham M 1985 Comparing software for the solution of systems of nonlinear algebraic equations arising in chemical engineering Computers and Chemical Engineering Vol 9 103 112 Shakhno S 2001 Some numerical methods for nonlinear least squares problems in Sym bolic Algebraic Methods and Verification Methods Alefeld G tz eds Springer Wien 235 249 Shampine L F 1980 Evaluation of a test set for stiff ODE solvers ACM Transactions on Mathematical Software Vol 7 No 4 409 420 Shampine L F 1994 Numerical Solution of Ordinary Differential Equations Chapman and Hall New York London Shampine L F Watts H A Davenport S M 1976 Solving nonstiff ordinary differential equations The state of the art SIAM Reviews Vol 18 376 411 Shampine L F Watts H A 1979 The art of writing a Runge Kutta code Applied Mathe matics and Computations Vol 5 93 121 Shampine L F Gordon M K 1975 Computer Solution of Ordinary Differential Equations The Initial Value Problem Freeman San Francisco 3l 472 Sheng Q Khalic A Q M 1999 A compound adaptive approach to degenerate nonlinear 473
153. COMP A Fortran code for automatic differentiation ACM Transactions on Mathematical Software Vol 21 No 3 233 266 Dobmann M Liepelt M Schittkowski K TraBl C 1995 PCOMP A Fortran code for au tomatic differentiation language description and user s guide Report Dept of Mathematics University of Bayreuth Germany Dobmann M Schittkowski K 1995 PDEFIT A Fortran code for constrained parameter estimation in partial differential equations user s guide Report Dept of Mathematics University of Bayreuth Germany Dolan E D Mor J 2001 Benchmarking optimization software with COPS Technical Report ANL MCS 246 Argonne National Laboratory Mathematics and Computer Science Division Argonne Illinois 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 Donaldson J R Schnabel R B 1987 Computational experience with confidence regions and confidence intervals for nonlinear least squares Technometrics Vol 29 67 82 Donat R Marquina A 1996 Capturing shock reflections An improved flux formula Jour nal on Computational Physics Vol 25 42 58 Donea J Huerta A 2003 Finite Element Methods for Flow Problems Wiley Dormand J R Prince P J 1981 High order embedded Runge Kutta formulae Journal on Computational Applied Mathematics Vol 7 67 75 Dorondicyn A A 1947 Asymptotic solution of the van der Pol equation Prikl Mat i
154. Data Fitting and Experimental Design in Dynamical Systems with EASY FIT Mek Pesign User s Guide Prof K Schittkowski Department of Computer Science University of Bayreuth Version 5 1 2009 Copyright 1997 2009 Klaus Schittkowski June 24 2009 EASY FIT 9 is an interactive software system to identify parameters in explicit model functions dynamical systems of equations Laplace transformations systems of or dinary differential equations differential algebraic equations or systems of one dimensional time dependent partial differential equations with or without algebraic equations Proceed ing from given experimental data i e observation times and measurements the minimum least squares distance of measured data from a fitting criterion is computed that depends on the solution of the dynamical system Moreover it is possible to predetermine an optimal experimental design by fixing the model parameters Additional design parameters for example initial concentrations or input feeds are used to minimize the size of confidence intervals Weight optimization helps to identify relevant time values where experiments can be taken The mathematical background of the numerical algorithms is described in Schittkowski 438 in form of a comprehensive textbook Also the outcome of numerical comparative performance evaluations is found there together with a chapter about numerical pitfalls testing the validity of models and
155. Equations Lecture Notes in Mathematics No 1402 Springer Berlin Carver M B 1978 Efficient integration over discontinuities in ordinary differential equation simulations Mathematics of Computer Simulations Vol 20 190 196 Cash J R Karp A H 1990 A variable order Runge Kutta method for Initial values prob lems with rapidly varying right hand sides ACM Transactions on Mathematical Software Vol 16 No 3 201 222 Chakravarthy S R Osher S 1984 High resolution schemes and the entropy condition SIAM Journal on Numerical Analysis Vol 21 No 5 955 984 Chakravarthy S R Osher S 1984 Very high order accurate TVD schemes ICASE Report No 84 44 Chakravarthy S R Osher S 1985 Computing with high resolution upwind schemes for hyperbolic equations Lectures in Applied Mathematics Vol 22 57 86 Springer Berlin Chang K S 1978 Second order computational methods for distributed parameter optimal control problems in Distributed Parameter Systems W H Ray D G Lainiotis eds Marcel Dekker New York Basel 47 134 Chartres B A Stepleman R S 1976 Convergence of linear multistep methods for differen tial equations with discontinuities Numerische Mathematik Vol 27 1 10 Chemburkar R M Morbidelli M Varma A 1986 Parametric sensitivity of a CSTR Chemical Engineering Science Vol 41 1647 Chen J 1991 Abk hlungsvorg nge von Stahlplatten mit Spritzwasserbeaufschlagung Um formtechnische
156. Example 2 12 LKIN RE We consider again Example 2 2 given by Y pu y1 0 p3 Ya pun Pey2 y 0 0 Experimental data are shown in Table 2 6 where we added artificial measurements with zero weights for being able to define constraints of the form 9 Pp 45 yo tj44 gt 0 for j 1 21 In other words we would like to limit the second state variable by the value 45 0 and require this condition at least at some discrete time values The differential equation is solved by an explicit integration algorithm with termination accu racy 1079 and internal numerical differentiation The least squares code DF NLP terminates after 10 iterations Constraint 13 becomes active and termination conditions are satisfied subject to a tolerance of 1077 Figure 2 15 shows the bounded state variable yo where y fits the data as in the unconstrained case 2 5 5 Shooting Method The traditional initial value approach discussed so far breaks down if we are unable to integrate the differential equation from initial time zero to the last experimental time t One possible reason is numerical instability of the ODE that prevents a successful integration over the total interval 28 1 1 2 NI t Yi tu V wi 1 1 85 34 1 10 86 1 2 2 79 44 1 21 1 1 3 3 79 13 1 27 01 1 4 4 71 18 1 34 45 1 5 5 96 2 1 36 41 1 6 6 0 0 0 0 0 0 7 7 0 0 0 0 0 0 8 8 0 0 0 0 0 0 9 9 0 0 0 0 0 0 10 10 29 66 1 46 27 1 11 11 0 0 0 0 0 0 12 12 0 0 0 0
157. Experimental Values In this simple situation the flux function is avoided easily by analytical differentiation by hand i e the dynamical equation is equivalent to e Dexp B c 1 czs DB exp B e 1 c Initial value is c p x 0 0 if x lt 0 and c p x 0 1 otherwise Dirichlet boundary values are c p 500 t 1 and c p 500 t 0 The remaining coefficients are supposed to be parameters to be estimated i e p D 8 T Experimental data are available for one time value t4 353 at 10 different spatial values see Table 2 10 When applying a fifth order difference formula with 21 lines DFNLP computes the solution D 627 0 8 3 608 within 26 iterations starting from D 400 and B 3 Maximum deviation of measurement values from experimental data is 10 26 The resulting surface plot is shoum in Figure 2 206 Another reason for using flux functions is to apply special upwind formulae in case of hyperbolic equations when usual approximation schemes break down Typical reason is the propagation of shocks over the integration interval enforced by non continuous initial and boundary conditions In most situations advection or transport equations are considered ut f lt p u gin Ween Z t d 2 88 where TOUR T Patha aa 0 t 9 aD t tas m t f lD 0 gs mt Example 2 20 BURGER_E We consider a very famous example now the so called viscous Burgers equation defined by the flux function flu 0
158. Fisher information matrix an eigenvalue eigenvector analysis is performed to identify significant parameter levels for subsequent elimination of non relevant parameters or further statistical design investigations Only for illustration purposes we denote the first independent model variable the time variable of the system the second one the concentration variable and the dependent data as measurement values of an experiment These words describe their probably most frequent usage in a practical situation On the other hand the terms may get any other meaning depending on the underlying application problem Due to the practical importance of parameter estimation very many numerical codes have been developed in the past and are distributed within software packages However there is no guarantee that a mathematical algorithm is capable to solve the problem we are interested in Possible traps preventing a solution in the desired way are e approximation of a local solution that is unacceptable e round off errors because of an inaccurate iterative solution of the dynamical system e narrow curved valleys where progress towards the solution is hard to achieve e very flat objective function in the neighborhood of a solution for example when there are large perturbations in measurement data e overdetermined models in case of too many model parameters to be estimated leading to infinitely many solution vectors e bad starting values for para
159. IE RISOUL y ye uoryenbo 310dsue yeoH uonrpuoo UOTPISULI YILM SLITE uonurs lur OMY uorjenbo eog Josueyoxe yeoy TUMANI Jojsuety yeoy eorrput A7 puno4bDsovq o SS n EH E cS o D O CC Ch Ch ooo CH CO Ch Ch Ch CH o ri CO CO CO CO Ch Oo o oo ooo Ch Ch Ch CC CH CO CC Ch CH vi CU CGN CGN NNN o rd rd HAN r r r GO Cl C ri r ri Cl r 06 06 06 06 86T GT 0 vL GE 0 0 GE OTT 66 42 0 OV IT Ic OT OV 0 0G m N CGN CV CGN CGN CH e Y MO 00 C Cl OC lei eo a CH VYOHHdAH OJUHAAH COS Hd AH LO HAH UaAdAH CNcdAH Odd dAH SOUDAH ATOd DAH XT OUGAH Se OYUAH O6 OHGAH OHGAH INNANH LOdS LOH X LVdH OGL LVAH XS LV4H NAS LVaH cS LVAH TS LVdH JIN LYH SIN LVI T LVdH Xd LVAH TAO LVHH IWDU 93 Sm gS gS PA vA Se gS gS SS SS uou gS gS IS gS gS gS gS SS 030p ponurjuoo FSI TGP TEG OFS OVS OVS SEZ 62 Troe as lezz Tezo OTT GES GES 839s GD OT Jo Surj3g snoouej nuis uoAo uorjooAuoo Y UL urrjxopoj eur JO SuLAIG uornjsodurooop YIM Sun ur uorjeor dde uro304q OUI UOISSTUISU 1 OLIJ2O O SSO SSO uro3s amp s orqde1goqeuroquo reout uON urojs s orqde1sojeuroriqo reouruoN pI9U Apo o qerreA HI uorjenbo orp oqiodKu reun 19p O 9S1 4 998 193U1 q3r4 uorjenbo orpoq1iodKu Ieun 19p O 9S1 4 uorjenbo o10q 19d y Ieou 19P IO 9SI1 H uor
160. IeQ IOAO 107CM MO eUS JO MOJA Krepunoq orporiod YIM SOTYISOISIA 9 Q9LIeA suorje ndod oA T oqn3 po e4 orj4 e392 V ur UOTJVOI I9pI0 OJI97 uornnqrnsrp yooys oue1quiour Aq poyeredos ses yya oqny uou uorjnpos 3oexo uorjenbo s 1og4ngp so ea Sut oAedT VIIL UOTPSULI YPM SOU IQUIOUI OA T uorsngrp peurriopsueiT uoryenbo spreypry toyem puno 3 jo uoryedjeg Xemusty e Som MOT otgre1T n uo1 Jop JO UOO uonnqrugsrp Year peurdo oung uorjenbo qdeiso oT Io e UOT ISUBIY UA SOURIQUIOU OM qsnorq uorsnpgip euLLopsue_Ly nbruuo 1 axn3d e or reur Aq poxoer ust WLIS U0Ij293op ssougng U0Ij293op ssougng punoAbDso2v0q om c CH CC CC Ch Ch CH Ch Ch Ch Ch Ch CH CO CC CH ooo Ch CH o Go CH CO Ch Ch vw OO Ch Ch CH ooo c Cl c ri ri vi vi ri DO r Cl 1 Cl Cl Cl Cl CH i ri ri rz 0G 09 001 O8T 0G OV OT 9 OFT gy qr GE 08 0 OT 9T S 8I GT TT Y OM ec 00 MLN DAN 00100 r ri CN CN Co e a cq os VHAVM EHAVM CHAVM THAVM YALVM SAODSIA SdOd OML Oda aL Aan AVA ANIL WHNSNVUL UAACSNVUL OAA NVHL OLLIV LL HNONOL LdO ANIL HAYA TAL SOJAAIG L HSIW ALS 6LACAALS TLAGAALS IWDU 99 10 7 Partial Differential Algebraic Equations Again we proceed from r data sets to yf E yl m E p where l time values and lr corresponding measurement values are defined together with weights wf Some of the weights can become zero in cases when the corresponding measurement
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164. L d 2 El 4 5 6 7 8 3 1 e 13 8 6 Delete Command The actual problem is deleted from the database All corresponding problem files especially the model function file with extension lt name gt FUN or lt name gt FOR are scratched from the problem directory of EASY FIT V c Design Alternatively a search mask may be defined by the user to delete a complete subset of parameter estimation problems from the database The search mask contains the following items Problem Identifier Model Name Project Number User Name Measurement Set Date The first five strings may contain a for determining a group of problems If a date is defined then all problems are deleted with date less than or equal to the given one Moreover it is possible to require confirmation of deletion of each individual problem Report Data Delete Make Utilities y i rimental Data Diffusion of water through soil convection and dispersion Demo Unit for Values f Schittkowski Unit for Y Values Narr Delete actual problem Select subset of problems to be deleted Figure 8 11 Delete Command 14 8 7 Make Command If model functions are implemented in form of a Fortran code the command allows to compile and generate executable code The corresponding compiler and linker calls are adapted through the Utilities command and must be contained in two DOS batch files with names COMPILE BAT and LINKE
165. L uorj29Auoo TIOS YINOITI 19I8M JO uorsngr q qes Jo o13uoo o1n3e1oduro Tepour uoryegedoud oureg suopueg 194M T uorsngrp eurropsueiT uorsngrp eurropsuei T urs 329 10d qy1m yuoturiedxa OINA ur Opou ung uis 329119d tu quourniodxo OINA ur Opou ung SOTJOULY uoremosse YPM opour ung sqy3tom pTeumjdo aam ye jgprqegnuoep L AALS uorngrp eusopsuery SI Ston q AALS uomngrp euttopsuedy Stop eug ye s eA1ogur oouopguoo ojenpeA G JH LS uorsngrp eurropsued YIM uUsIsop ejuourLiodx q puno4bDsovq oo Ch Ch Ch CH CO ooo Go o T oo OGOGO SO 00 ed LO eh e e eh CO CC CC O ri rd Mr rz MA CN CGN CN CO TAN r G OTI 0 I8 0 9I OT IV GG IT Ll 0 VI 08 OT 9 GC 9G 89 GC GC GC SC NN DON ON oD CO Go GU CH CH N om omom o NVHdHLS dOLYUVLS CHA Ld VILS LAN YVLS dUdHd INDIOS NOLLd HOS EST duos CSI dHOS IST duos NOLI IOS qrios TIOS ULO qV IS avs SNIMS VNDIS X ENIMS ENDIS GNIMS 9INDIS SINDIS VINIMS IWDU 58 IS SS 0S IS GS GS gS 030p eer use vea Oct gzz lost cT TIS e gos oor Jare lesz FG ber Ic per per fas Ulo s s or oqiod amp qg rures suorjoourp ayisoddo ur Sur oAer SOARM OM uoryenbo osem ot oq 1od App suorjenbo oroquadAy oA1 JO woz ur uorjenbo 9AeAN u mouy OLMOS 3oexo uorjenbo oses 2000 1900 JOLL
166. MO c Cl CGN DN N A N cO co Re NAC OHTA OLLSV TId ADO IOOH AUC LHHSIG HUSIA TAAAVSIA YOLSHADIA SOHATO IVANA Vol IG FLdddIO SLd44IGd GLddAIG LLd4A4Id Ld4A4AId d 4A4AIG q IN HATO HLY 441A NOO 441A Sav 44IG OT A34IG SISA IVIG 9 Du 50 gS gS gS gS gS gS gS gS gS gS gS gS gS SS SS SS SS SS SS GS gS GS gS GS GS GS GS SS 030p ponurjuoo Pr 66G G6 666 66 666 666 666 C66 C66 C66 666 c66 C66 666 D 666 666 66 G6 G6 G6 666 66 666 VET AAT 08v ser fas BUSEM 1041 T 10 uonnIos eqo 8 ou ut qo d gott wo qoid uorsngt T reour uro qo id onyea TO oroq iod Aq wo qoid uorsngt T uro qoud uorsngrp re ut uoN uro qo1d uorsngrp onpeA amp repunoq Tqerjrug uro qoid Goar YOOYS ouo OJU SIS IO SYDOYS OAM YPM urorqo1d onpeA eru yyed 39078 Ota uro qod onpeA Tout 9A A JO UOTZBULIOF uro qoad onpeA erjyrug uro qo1d onpeA amp repunoq Tqerjrug SARM gup eo1q uro qo id onpeA erjrug UOTVULIOF YOOYS YIM uro qodd n A erjrug uiro qo1d on eA erjtur TeourpuoN uio qo1d on eA erjtur TeourpuoN uro qoad on eA pertu uio qo1d on eA erjtur TeourpuoN uro qoad on eA pertu uro qoad on eA pertu uro qoad on eA pertu 9A A BUTTOARLT uio qoad onpeA amp repunoq Terarug on eA VIPUL snonurquoo uou YIM uro qo1d onpea PINUL uro qoud Surpeustg uorjoeor oryeur amp zuo pue uorsngrp orureuA q uoryenba A
167. Mathematical Software Vol 20 No 1 63 99 Peters N Warnatz J eds 1982 Numerical Methods in Laminar Flame Propagation Notes on Numerical Fluid Dynamics Vol 6 Vieweg Braunschweig Petzold L R 1982 A description of DASSL A differential algebraic system solver in Proceedings of the 10th IMACS World Congress Montreal Canada Pfeiffer B M Marquardt W 1996 Symbolic semi discretization of partial differential equa tion systems Mathematics and Computers in Simulation Vol 42 617 628 25 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 Pfleiderer J Reiter J 1991 Biplicit numerical integration of partial differential equations with the transversal method of lines Report No 279 DFG SPP Anwendungsbezogene Opti mierung und Steuerung Technical University Dept of Mathematics Munich Pin Gao Gu E T Vishniac J K Cannizo 2000 Thermal equilibrium curves and turbulent mixing in Keplerian accretion disks The Astrophysical Journal Vol 534 380 397 Pinter J D 1995 Global Optimization in Action Kluwer Academic Publishers Dordrecht Boston London Plusquellec Y Courbon F Nogarede S Houin G 1998 Consequence of equal absorption distribution and or elimination rate constants Report UFR de Mathematiques Universite Paul Sabatier Toulouse Poeppe C Pelliciari C Bachmann K 1979 Computer analysis of Feulgen hydro
168. Meh Vol 11 313 328 Translations AMS Ser 1 Vol 4 1 23 Draper N R Smith H 1981 Applied Regression Analysis John Wiley New York DuChateau P 1995 An introduction to inverse problems in partial differential equations for engineers physicists and mathematicians a tutorial in Proceedings of the Workshop on Parameter Identification and Inverse Problems in Hydrology Geology and Ecology J Gottlieb P DuChateau eds Kluwer Academic Publishers Dordrecht Boston London 3 50 Dunn I J Heinzle E Ingham J Prenosil J E 1992 Biological Reaction Engineering VCH Weinheim Dwyer H A Sanders B R 1978 Numerical modeling of unsteady flame propagation Acta Astronautica Vol 5 1171 1184 Edgar T F Himmelblau D M 1988 Optimization of Chemical Processes McGraw Hill New York Edgar T F Lapidus L 1972 The computation of optimal singular bang bang control II Nonlinear systems AIChE Journal Vol 18 780 785 Edsberg L Wedin P A 1995 Numerical tools for parameter estimation in ODE systems Optimization Methods and Software Vol 6 193 218 Ehrig R Nowak U Oeverdieck L Deuflhard P 1999 Advanced extrapolation methods for large scale differential algebraic problems in High Performance Scientific and Engineering Computing H J Bungartz F Durst and Chr Zenger eds Lecture Notes in Computa tional Science and Engineering Springer Vol 8 233 244 Eich Soellner E F hrer C 199
169. NA ION MVLOVUA LSQHO4 SONIC TOA PONIA TOA ONIC TOA CONIC TOA TONIC TOW ISAYON TA SHONA AONTA TO CIO TA dOd HSIA NIA IWDU 28 Dono a q soc es oci eS orc es ror os see os see q es rez IS vec eS gor IS rec a es rec es gor es roc eS 661 eS 661 X IS pez q s es oec os cet es Lo q 6sz es oec q str 030p fo uorjenbo eruatopip 913s130 SURIQUISTI Y3N01Y Moden WOT sonou SUIPUIG T UBNA I9UT J9JU9ULIS SHONUTJUOY V ur SOTURBU AP u S xo prnbi pue sex SSQUJIIS SUISPIIDUL UJIM sur qoid IS99 JO Sept eryuo3od oarou jo s ndur D SAOIA9H I9UIJOA JO uorjn ossrp Jegou Aq uorjonpold ser Tepour o qegruopr ATTGOOUS A pem39n138 1039891 Mop Sn d 03 uoryeumrxoadde 1 1SO OPLIPAGUE 919998 Jo SIS TOIPAY 1039891 qojeq UOI BZIIQUI TOA Rorpel 99 1 sno u souro o ou ALM o durexo I897 oruropeoy 10j2 13X9 poyey13e ue ur so goud dnp ou quorsuem sjodysep pue sSurids re uruou yya sorjodoud 81199801 quouresrT oururerrouo Aq3ourexou JO o1njoegjnueur qojeq ruiog uonngo4o oruoprdo SATV ATH uq WOI JO JURINU VIW Y 0 pue urquiodq3 0 Surputq UIPNIH sisAyeue Ajr10L1d Jop ansst qued Jo uorjerquo19grp pue q34or 431 Aq ooue IPBLIL JO S 9A9 YSTY ye onssty gued jo uorgerguo19grp pue q140 1 4 Jop1o Jo uorjenbo erquo1ogrp Areurp4 1ogueuoxo yeoq 9QNI PUR 9YS Jo sorureuA T urojs4s Krepunoq 3urod oA urojs4s ueruoj rurerq odooso1KS orrjoururAs
170. NAWHOIN THON OTHON 60HOW O LAMYVIN LODTHVIN OOW OVN X NDIT X NINT CINOIT SIA NIT GOIN NIT NIM NIT X OH NTI dIND NTI CAWO NTI IdIND NIT IWDU p nuruoo a a gS uou uou uou x gog uou T og tos coe 00c 9uou 9uou 9uou 020p zot 608 c9G c9G 0F 0F UE UE UE UE UE GGE foa ourprumb jo sorjoup ooeurreud uorje ndoq Tepour o qe3s amp repuooos pue Areuirtq ouroqos 19 01 JsATeyeo Suisn uex q ur ouorpegnqATqod sto y3ty jo uorezrrourAqoq 1o3 g uorjezrre od jo oouooso1on q SON RA SUIT UI SIO LIO YPM FUNYY jep perurouAqToq s 910 erxe Surjnduroo 10 uorjeurrxoddde erurouAqoq T 939q Dt uro qod s9 uoryezrumdo eqo g 0 3 q p u uro qo1d 9593 uoryezrumdo eqo o 0s e39q p u ut qo d 9893 uoryezrurmdo peqo o sonpeA Te3uourjodxo pojnqunsrp AT VULIOU 09 opour uorjeogrjuopr 1ojourereq sonpeA Te3ueurj dxo pojnqunsrp qpeutrou QE opour uorjeogrjuopr 1ojourereq sonpeA Te3uourjdxo pojnqunsrp euniou GT opour uorjeogrjuopr 1ojourereq sonpeA Te3uournjodxo pojnqumsrp AT RULIOU QZ opour uoryeogrjuopr 1ojouredeq uro3s As Surye Ds uorjn os UMOUY JOVXO YIM uro3sAs Surje s uro qoad o1quoo eurrjdo sour Surqojras o qerreA YALA Soroupr reourT j 0 JO sseur otqq Syeo JO sseur otqq sjure1jsuoo ueur AIOA YIM SuIeISOId re ur uou SurjsoT sjure1jsuoo Aue AIOA UM sure1do1d 1eo
171. NLP has been replaced by the codes NLPLSQ least squares data fitting NLPLSX least squares data fitting for very many measurements NLPL1 L data fitting sum of absolute residual values NLPINF JL data fitting maximum of absolute residual values Disclaimer THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIB UTORS AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FIT NESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEQUENTIAL DAMAGES INCLUD ING BUT NOT LIMITED TO PROCUREMENT OF SUBSTITUTE GOODS OR SER VICES LOSS OF USE DATA OR PROFITS OR BUSINESS INTERRUPTION HOW EVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARIS ING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE Contents DI Wetec eke ede W E S Oe ee Eee BAe ES REE Sa se s i 0 1 1 Hardware and Software Requirements i ULA Pagane List sos ee ee vo xoxo REE PAG Oe PER Oe FA SD Ge x i DIA System SetuP cou em s ew s S S w ESOS ie jii 01A Starting BASY PI errr ng ag gpa ed bra sqa s s nu v 0 1 5 Dimensioning Parameters 2 E 39x Rx OX 43 RR ES vi 1 Introduction 1 2 Data Fitting Models 1 21 IMP edo ns s eb S
172. NOI INVULNOI JOUT ANI CAYHALNI TAYHALNI DALNI UAAV T NI SQdYUAdAH IWDU 54 IS IS IS SS SS TS X gS gS gS gS gS gS SOS gS gS IS IS 0S gS gS gS gS pop Dono ere oer eer OFT 066 er GCE Oct la Icy ITS 9z8 Tel KZ ggg oeg SP eer eer TG e2 686 Scy vst Add oyoqsedAy Jo worynjos A1tole lIoso 1 qumu Start 3run uou pue UOISNYIp YIM uorjoeor days UN 9 98ISGO UL IOAO MOF Jam MOTTeYS uorjenbo uorsngrp uornoApe Ieoui uON e ouro1 stou pue Surr m qop reouruoN xo duroo uoranqos uogos Joex uorjenbo 1o8urpooauog reour UON uornj os j2exo YHA SAAd Ieourmuou OML SOATJVALIOP erj red 1opio puooos reouti uON uorjdaos jurod 0M3 reour uoN Arepunoq orpo uod 193Mg yooys e Surdo oAop uoryenbo oda reouruoN UOIjn OS 3o XO YPM AC Ieour uou IOplO ISA uonnjos 32exo YIM ICT Ieourpuou AIS uorjenbo yeoy reoui uON osmd SALON S IOS 59 103 ur sorureu amp p umnruourure pue uoSod4tN xng xoAuoouou IYSO pu nyg jo uro qo1d 3893 or oqrod AP SYMONS P9ye 1393ut jo uorjonpodd 10 ooeuunj eorrjoo o SUOZIANDIAN uoryenba uorsngrp uorgooApe poqgour Adog3uo urnuirxepq uoryenbo s 193Mg 3uo3j SULAOJA wo qoxd onpeA repunoq uotsngrp Ie nasToNx syu rouJ oo5 I9FSULIY Jey Suryeurrs SUI OOD rur BUOY uomnrpuoo amp repunoq ut tyd o qerreA Buroo ru Suoy uonrpuoo Arepunoq ut rsd 3ue3suoo SUI OOD prua Sujoy
173. NPDE 2 NDIS NPDE NDIS IF THE SIZE OF THE LAST 4 PARAMETERS IS TOO SMALL PDEFIT WILL REPORT AN ERROR MESSAGE pis IMPORTANT DO NEVER CHANGE THE COMMON STATEMENTS OR THE CONTENT OF COMMON VARIABLES UNLESS YOU KNOW PRECISELY WHERE THE VARIABLES ARE USED IMPLICIT NONE INTEGER MAXCPL MAXPAR MAXPDE MAXODE MAXMEA MAXDIS MAXTIM MAXOBS MAXRES MAXCPB MAXPOI MAXPLT MAXDDP MAXBPS ie LVA1 LWA2 LVA3 LKWA LWLOG DEMO MAXDAE LFLUXP LMAXLS QOOO0OOOOOOOO0O00000000000 DIMENSIONS ann PARAMETER MAXCPL 100 MAXPAR 51 MAXPDE 50 MAXODE 2000 MAXMEA 30 MAXDIS 1001 MAXTIM 500 MAXOBS 2000 MAXRES 200 MAXCPB 50 MAXPOI 10 MAXPLT 200 MAXBPS 100 1141 3500000 1W42 1000000 LWA3 10000000 IXVWA 50000 IWLOG 100000 MAXDDP 1000000 MAXDAE 500 LFLUXP 100000 IMAXIS 500 LOGICAL PLTEVL INTEGER NMEA NODE NTIME NBAND NOBS NPAR NPARO NRES NEQU NPLOT NCPL NCPLO NCPLA NPDE NDIS NCPE NCPB2 NBOUND ITIME INUGRA IPRINT NOFUNC NOGRAD NOMODF NOHODG TERR ICOUNT DQUMET MAXPDE 2 MAXCPB 2 DQUPOI OPTMET IOPTP1 IOPTP2 IOPTP3 IODEP1 IODEP2 IODEP3 INTEG ICOMP MAXCPB 2 LBOUND MAXPDE MAXCPB 2 REOUND MAXPDE MAXCPB 2 IBND MAXCPB IXBND MAXCPB zi wat REE AE S Figure 8 16 Dimensioning Parameters 20 System Configuration Data x System Directory C Easyfit E asy fit mdb E ditor JEDITOR EXE Graphics System EASYFIT C I O Fortran Compiler INTEL Cj BEEN Default Export Import Pat
174. O oS8esop PIJOS JO asvoTaI ojerpourui uornnqrnmsrp eurrouso SULIO 98esop ptlos JO se 1 ojerpouruiT UOTJNLIYSIP 9393ostp SULIO oS8esop PIOS JO se 1 ojerpourui JOpoul p pu x SUIIO ogesop PIJOS Jo oseo o1 ogerpouruiT uor prxododns JO uorjoe p zt e1eo ZO pu OZH 03 uot prxosodns jo uorjegnuist T INJ NS VATPOROIPeY Sossv OUL uo mm sno nutjuoo IO yoyeq ur Si riqns JO UOIjeurio jonpo d pue YIMOIL s oso1oe aseyd 0M3 YPM uorjep numooe 93e 1ISqng utr qo d 3893 HS ut qo d 3893 HS ado Jus ado Jus uoryenbe erquod9gprp Jug GLV LS JO uorjer amp ioudsoud Aq 10suoo o30uro1 e se ZUIL orusejdog450o on N uro qo1d poq T euorjejrAe 1o Kxepes peorrpui Ao e jo erjyuo3od oY UYM 1ejs e Jo UOLJOT uorniqrnqui JONPOIA yya oan3 no 938IS OM T sooueqnjsrp YPM s1032eo1 eoruroqo Jo AATTIGeIS JUL porrms snonurquoo Ieouti uou SUIZIILIS Jost yeoy ssed omy aye4s Apeays ayeys Apvoys oqn e ur uorjo oAuoo uoIsngt T 103e 1950 oruourreq poorogun Suds reourT Tepour Surxrur 1039891 paq poynods Squouirodns eour ueures pue 10J seurep ouo Aqjo pue ueuj ur Surjoog uornjos p eorjAqpeue ILM o durexo eurg punosbyo0q o ooo oo CH oo oo co CC CH oo ei OH Ch Ch CC Ch CH o OC CC oo CH ri CEO CO Ch CO CC CH oo ei OH Ch Ch CC Ch CH MO ri rd rd r r r ri O c2 CN GO Cl Go P Oo e O N WV GO QV CGN CO F CH 86 86 86 6 v6 86T FOI TE 06 Ic
175. OIJOUL 31q10 o duutg PART oTIyHode oy uoAIS 3 sosop pue uorsngur Tt jo uorjdope Tod Soje3s euy DOT YILM o1juoo ewn do Sjure1jsuoo 0324S Iop1o pug YILM uro qo1d o1juoo feudo punosbyo0q ooo CC OCH Ch Ch O C Ch O Cu Ch CH Ch Ch CO Ch Ch CH t O ooo G UOU OCH QT Q O CC CH Cu Ch CC Ch Ch Ch Ch CH N STN Y OC co C aO lt lt co CO c eo Y N NAAA TODO co co co CH g mo A GE GE 0c OV 08 0 RU 06 8C 0 OV VG 6c STI OST VIE 98 GS VG OV 09 6I 6c OM C MID MAIO N CN AN Na ei ATT Oo c o lt NAT VHd INAG VHd CIOLLSdd IIDINHd OI ONT VIT Nd NVOUd SMVdd god ATOLLYVd AZIS HVd H4NOZO LVGIXO ZNAXO NAGXO IIDSO TIOSO WINIGOSO NVUL OSO OVA OSO ODAYO LITO NLOW duo NIM LdO LO LdO INOD LdO IWDU 34 Dono gS a gS a a a IS IS SS SS a 0S gS a gS a gS a IS 22 04 9c 1 vor Jeng zs ret oor 9c 1 fas syunoo uorye ndoq UOLYBZLIQUI TO Y jsKpeyeo Sursn oO ur ouorpejnqATod sto qsrq jo uorjezrrourAToq UOLYBZLIQUI TO Y UOLYBZLIQUI TO Y jue3n od jo yuatuyea1 1039891 Ie nqnj mopy 3n1 4 umurje d uo uorjeprxo OD surstueS100Jorur jueurquiooo1 JO APLIS senpeA VIJIUT JO 39891 YIMOI3 JUE Y KytArjonpuooojord juersuey ISBT sso901d srsouquAso30uq sque d jo uorgonpoado3oud Aeq ssooo1d SIS9YJUASOJ0UY dq oouoosoroqdsorud uorjoeor LIWY uro3sKs 19Jsue130ud
176. ON TOJON LSIQ OW HAC TACM CV LVaH GROEN TIN LVAH O IN LVAH A LVAH V LVdH AAI MO TA WAN LVIA IWDU 63 Bibliography 1 2 10 11 Abbott M B Minns A W 1998 Computational Hydraulics Ashgate Aldershot Abel O Helbig A Marquardt W 1997 Optimization approaches to control integrated de sign of industrial batch reactors Technical Report LPT 1997 20 Lehrstuhl f r Prozesstech nik RWTH Aachen Adjerid S Flaherty J E 1986 A moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations SIAM Journal on Numerical Analysis Vol 23 778 796 Ahmed N U Teo K L 1981 Optimal Control of Distributed Parameter Systems Elsevier Amsterdam Alonso A A Banga J R Balsa Canto J R 2002 Model reduction of complex food pro cesses with applications in control and optimization in Computational Techniques in Food Engineering Editorial CIMNE UPC Universidad Politecnica de Cataluna Barcelona ISBN 84 95999 13 7 155 169 Anderson D H 1983 Compartmental Modeling and Tracer Kinetics Lecture Notes in Biomathematics Vol 50 Springer Berlin Anderson E Bai Z Bischof C Blackford S Demmel J Dongarra J Du Croz J Green baum A Hammarling S McKenney A Sorensen D 1999 LAPACK Users Guide Third Edition SIAM Philadelphia Andersson F Olsson B eds 1985 Lake G dsj n An Acid Forest L
177. Otherwise these lines contain the following data in un formatted form for 1 to NTIME ti i th measurement time not smaller than zero yf wE measured data i e experimental output and indisid sl weight factor for EE with number k k 1 NMEAS Note that the time values must increase In case of NCONC gt 0 the concentration values must increase as well and the set of time values must be repeated for each concentration 37 a6 4x i5 NLPIP Output flag for NLPQLP when computing consistent initial values in case of a DAE or solving system of nonlinear equations in case of steady state system NLPIP 0 no output at all NLPIP 1 only final summary NLPIP 2 one line per iteration NLPIP 3 detailed output per iteration a6 4x 15 Ree Maximum number of iterations for NLPQLP when usw consistent initial values in case of DAE a6 4x g10 4 NLPAC Final termination accuracy for for NLPQLP when com puting consistent initial values in case of DAE a6 4x 15 Number of constant break points where integration is restarted with initial tolerances Constant break points are permitted only if NBPV 0 and ODEP1 lt 7 Unformatted input of NBPC rows each containing one time value in increasing order that represents a break point of the right hand side of an ODE DAE Among the data generated by MODFIT are result report and plot files that can be used for external programs evaluating these data The format is described
178. R BAT Report Data Delete Make Utilities rile ae Model D e TIT Link rimental Data PDE Diffusion of water through soil convection and dispersion Demo Unit for Yalues ft Schittkowski f Unit for YValues Narr Figure 8 12 Make Command 15 8 8 Utilities Command Through a couple of menu items EASY FIT 9 can be adapted to individual situa tions The following subcommands are available Report Data Delete Make Utilities r r I Mo del Desig Compiler options Linker options Dimensioning parameters rimental Data System configuration Time values Generation of measurements Diffusion of water through soil convection and dispersion Demo Unit for X alues ft Schittkowski Unit For Y alues Narr Figure 8 13 Utilities Command Compiler Options A DOS batch file with name COMPILE BAT must contain all necessary compiler options Default execution commands for various Fortran compilers are included and can be modified in the editor window displayed Linker Options Similar to the compiler also all linker options can be adapted and reset by a user The file to be edited is called LINKER BAT The object codes required to link the complete program are set automatically by EASY FIT ModelDesign Dimensioning Parameters Parameter estimation problems can differ in their size dramat ically In some cases we have an extremely large number of exper
179. SOFT F77 Compiler rem Figure 8 14 Compiler Options System Configuration EASY FIT c Pesi needs to know where to find some files Thus a couple of directory names must be set according to the special needs of the user and depending on the special environment Also an alternative Windows editor may be defined to be used for the input of model functions It is recommended to check the available strings immediately during the first installation of EASY FIT 9 The actual version of EASY FIT s 9 possesses interfaces for the Watcom F77 386 the Salford FTN77 the Lahey F77L EM 32 the Compaq Visual Fortran Absoft Pro Fortran the Microsoft Fortran PowerStation and the Intel Visual Fortran compilers The corresponding compiler name some path names and the execution commands for compiler and linker are set in a configuration form Moreover the graphics system may be changed see the description of the Report com mand for more information In addition the form contains flags for editing GNUPLOT commands before executing the plot program and for allowing overlay of plots in case of us ing the standard graphics A default path may be set that is inserted into the corresponding Tf form when looking for files for importing or exporting numerical data and model functions Generation of Time Values The generation of equidistant or exponential time values might be the first step to generate a data fitting problem Also for
180. Schriften Vol 30 89 90 91 92 93 94 95 96 99 100 101 102 103 104 105 Chen G Mills W H 1981 Finite elements and terminal penalization for quadratic cost op timal control problems governed by ordinary differential equations SIAM Journal on Control and Optimization Vol 19 744 764 Cherruault Y 1986 Explicit and numerical methods for finding optimal therapeutics Mathematical Modelling Vol 7 173 183 Chicone C 1999 Ordinary Differential Equations with Applications Springer New York Chudej K Petzet V Scherdel S Pesch H J Schittkowski K Heidebrecht P Sundmacher K 2003 Numerical simulation of a 1D model of a molten carbonate fuel cell PAMM Proceedings of Applied Mathematics and Mechanics Vol 3 521 522 Clark C 1976 Mathematical Bioeconomics Wiley Intersciences New York Collatz L 1960 The Numerical Treatment of Differential Equations Springer Berlin Collin R E 1991 Field Theory of Guided Waves IEEE Press New York Colombeau J F Le Roux 1986 Numerical techniques in elastoplasticity in Nonlinear Hyperbolic Problems C Carasso P A Raviart D Serre eds Lecture Notes in Mathematics No 1270 Springer Berlin Crank J 1970 The Mathematics of Diffusion Oxford at the Clarendon Press Cunge J A Holly F M 1980 Practical Aspects of Computational River Hydraulics Pitman Boston Cuthrell J E Biegler L T 1989
181. Sious uo poseq 1039891 re nqn uro qo4d 9599 20 opour orureu amp pod359 4 puno4bDsovq O O O CC Ch Ch Ch CC Ch CC Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch Ch CH T CC CC Ch CH Ch CH Ch Ch CC CC Ch Ch Ch CC Ch Ch Ch Ch Ch Ch Ch Ch CH Y N rd Cl rel ri ri Cl cl rel el rel rel rel ri ri ri ri ri rd Cl gt amp 0 OT OT 8 OT OT OT OT VG 8C OT IT IT OT 0 SI OT OT 001 0 VI SI OT GG cv 8E HSVMUTIA q GO Xu V V9uxM q SX EVAXA EVuxd GVuxd cl Ves Xu 6 cEUXH 8 CE YU XA L CEUXH V ceux TGeuxd VISWXMH 9 GCHXMW SCUXad TS uxa LTGUXMH TecuXa TG cuxd LICHXMH V STHXM STIWXH FIN LUTIWXMH NAGZNMW ADUANA OILLdI TTA OYLOTTA IWDU 51 SS SS SS SS SS gS gS gS IS gS E mE gS gS gS IS gS gS IS SS IS IS gS gS 030p ponurjuoo cer eer 1c eet loz ITG Jore c1 spz ICI LLE LVE IGP LTS LTS FIG Hat 918 26 SPE fas xnp e 8M 3u 3suoo YIM uro qO 1d z3oeirc uomo Auoo Aq SSOT YOY YIM Iopul AO ur I9Jsu er Je9H UY Ie NIALID Y ur IoJsue1j JOLT uornonpuoo 3eoq euorsuouip ou oda pity jo suorrpuoo amp repunoq uoryenba yeoy reourpuoN Uornjrpuoo uorjs UBI PALM seore uoryerdojur OM pue squIod eo1q uorjenbo yeoH uorjenbo eog uoryenbo s 19YSIJ u34o18 uorjemdod jo opour 91981380T uoryenbo sp reqo
182. Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the relative global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible It is recommended to start with a relatively large accuracy a low number of iterations and to increase the accuracy when approaching a solution by restarts Initial Stepsize for Solving Differential Equations Define initial stepsize for differential equa tion method used The parameter is adapted rapidly by internal steplength calculation Bandwidth of Jacobian of Right Hand Side Implicit methods require the evaluation of the Jacobian of the right hand side of the discretized system of ordinary differential equations with respect to the system parameters since they have to apply Newtons method to solve certain systems of nonlinear equations The Jacobian is evaluated either numerically by the automatic differentiation features of PCOMP or must be provided by the user in form of Fortran statements In all cases it is possible that the Jacobian possesses a band struc ture depending on the discretization scheme EASY FTTModelDesign allows to define the bandwidth that is passed to the numerical integration routines to solve systems of internal nonlinear equations more efficiently The bandwidth is the maximum number of non zero entries below and above a diagonal entry of the Jacobian
183. Stot yya ooeuang ouung Doug ur Gott eIpaul sno1od ur MOP uoALrp o1nsso1q SUOIJ99 9 PUL suor jo 3uouroAout outrj ooedg uorjn os 32exo YIM TV Td I9p1o YANO o durtg ULIO SNUIS snoouoSourouut YHA HUA t oqereq jOUY UUBUMOIN PaA OIJUOY pue sure q 99119 YIM YIOMION 1199 Ion OWON 1199 PM OJON 1199 Tong OJON 1199 ong OJON uornjernsrp 3uouoduroo rjn JA suojr os oonpoad oi uors1odsrp pue Suruodaeqs quod Surue eq MOP 1996A AO TeqS UOJIOS ouo yya uorjnjos 3oexo uorjenbo soLrA 9p 9949310 Y UorjooAuoo PIDIO pue uonrpei YPM uorjonpuoo 169H uon enbo ore1qoS e pue suorjorijsor eorureuAp yya uoryenbo yeoH d uueuizj ogq uejojg jo uonrpuoo amp repunoq reouruou ujr uorjenbo map od uueuirzjpog uejojg jo uonrpuoo amp repunoq reouruou Ota uorjenbo map UOIj nULIOJ xnpg pue uoryenba oreriqoS e tw poje nurio uorjenbo map uorjenbo ore1qoS e yy p le nurioj uoryenba yeoH uorsngtp YM urojsAs MOJA ou Iquiour jeg I9AO Io e repunoq uorje1juo2uo puno4bDsovq T CC CC Ch Ch coo Ch oo CC CO c CO Ch Ch CH Y CO CC Qh Gu CH On Ch Ch CC CC CO CH NN O O NA Cl CV Cl Cl c c ri Cd c ri ri 0 n ri ANN A cO CN CN DOM vi Cl NN OT VG RU qI 0c v8 001 OL OTI OTI 0 9 LT vr OT LG OT OT LG LG TOT 0c N co i ON CN GO DOIN NO c N Glo XA CH WVdd HIA WVdd INA THANNAL NOOTIIS AYASSAYUA VINSV Id HV d VNIS HVd LAN ONON CMN GOJ
184. T 0G OT 9 9 9 82 9 ve 9 o CH CN F ON O m o o ri N NA NN si CN OM GO XH CO XMOUDDOIN MOUDOIN TAHLAIN NVHLAN ANAS LAW SAANEN dASWAN HNIWAN SAS HOUN XTILLSON TILSIN HNTUVIN OHdINAT ION LOT TIOA LOT SZNAYOT ZNAYOT MOUD DOT INOW ODOT LNDII S NDIT awd NDIT O NIT WON NINT VTNINI FV NIMT 9t Du 3l p nuruoo a IS q a a sog q a SS SS IS gS a uou gS 0S 0S IS a a a es 6ze dd CH 030p 80S vez 971 ve QNO vec IT ot 068 us ey er ees N Ze prxo U9S30 JTU JO UOLT oseud ses SNOUISOVIOY 9 ISIAH our rue 0j ouozuoqozjmu JO UOISI9AUO 193SUe 1 USS XO YPM UOVU yore opis PUBY 17311 snonurjuoo uoN uoryeuoSorp u 19989 ATOPY S 99 o osnur ego os ur oseo o1 J ce7 998J1MS orregour e oo so noo our jo uorjd iosop uorjd iospe 10 opour Jop uone msrp Omg 3ueuoduroorg n jq LUIL V ur 189 V JO uoropv uro qo1d ueurso es BUNALI pozrrojour JO orquoo peurmd wo qoid drqsooeds uoour qj3 reo euorsuouip ou sojep nue1S Jo 9IMISIOTA uro qo id ox3uoo feudo urrou urnuturjA Uuorjoe ourAzuo JO SOLJOUTY SOLJOUTY uoquojA si oeqorqN eursepd pue ourm ur ssoooad or oqe39 A SOLJOUTY USJUIA SI 9CYIT A eursepd pue ourm ur ssooo4d or oqe39 A sorureu Ap uorje pndod amp oad 10jepoaq OP 103 10 otqno uorgjoeor eoruroqo TOPOUI 9781 p xt UOV eoruroqo t Topour 91 PIXTIN
185. TVD schemes Lectures in Applied Mathematics Vol 22 381 395 Springer Berlin Zachmanoglou E C Thoe D W 1986 Introduction to Partial Differential Equations with Applications Dover Zeeman E C 1972 Differential equations for the haertbeat and nerve impulse in Towards a Theoretical Biology C H Waddington ed Edinburgh University Press Vol 4 8 67 Zegeling P A Verwer J G van Eijkeren J C H 1992 Application of a moving grid method to a class of 1d brine transport problems in porous media International Journal for Numerical Methods in Fluids Vol 15 175 191 Zhengfeng L Osborne M R Prvan T 2002 Parameter estimation of ordinary differential equations to appear IMA Journal of Numerical Analysis 37 562 Zschieschang T Dresig H 1998 Zur Zeit Frequenz Analyse von Schwingungen in Antrieben von Verarbeitungsmaschinen Fortschrittsberichte VDI Nr 1416 VDI D sseldorf 489 506 38
186. Table 2 11 case 3 In all three cases the differential equation is integrated by an implicit method with absolute and relative termination tolerance 10 The example shows that the parameter estimation problem is solved more efficiently and more accurately when introducing switching times There is however a drawback of the proposed approach when too many switching times are defined If some of them are redundant and become identical at an optimal solution as for case 3 of Example 2 10 then the final optimal solution is not unique and a slow final convergence speed must be expected 23 Dry Friction u 0 01 12 x1 t X t 6 t Figure 2 10 Function Plot for x1 t and x t Dry Friction j4 0 01 0 5 t Figure 2 11 Function Plot for t t and a t 24 Dry Friction u 1 5 1 6 72 7 z1 t z t e NO R o 00 Figure 2 12 Function Plot for zi t and x2 t with Two Switching Times Dry Friction u 1 5 4 6 T2 7 Zoll taft 0 Figure 2 13 Function Plot for t t and t2 t with Two Switching Times 2 5 4 Constraints Constraints in equality or inequality form g p 0 j 1 iig 2 50 g p 20 j me 1 Mr 2 90 can be added to the general objective functions 2 1 2 2 and 2 3 or to the data fitt
187. Teous pum 1opun sorio3oofer jo oxe3 pudo uorgenbo oruropeoe p uonrpuoo uorjgoduioo UL spooA om JO 140117 uueumoN pezudostp Add uomnn os joexo YIM uorjenbo sosem ot oq 9d T Prend p zr osip Add uonn os joexo YIM uorjenbo oA A stjoqIiod H Quourjeor JOJVMOISEM orqodo uV HOT oyerjsqns Uu opour pouou juourjeolj Io3eM93jS A IIGOIOVUY RIP MMOYIM opour pouour 3uourjeo1 19JLMIJSOM oIqooeUV uorjd ospe 9SSIA dA 1op1o qp ouueqo peorydoA Buoj Y UL Mog pmp poyoluy m peonrjoo o uorjenbo Og 19p UBA m PeZ uorjenbo oq Joep UBA uorjequourlo oum oA ALLA 1039891 Ie nqn oseud ses WOT VIYWIOUOD ojroqejour Sur amp reA YIM opour orurgu p eorur uoorq IeoUTTUON ALSO Y Ul uorjoeo1 x duroo e Jo o3e3s Apeoysuy punosbya0q o ooo CH oo ooo Ch o Ch Cc CH qu oo ooo ov co Cc CH o qu N cO ao Co CN co CV CN YANAN 00 0 qI SI 9I 001 001 VI 0G ST 0 8I 9T FI 84 vOS GL OTA CN GOO CN YT y CO CO MNV IdOOZ OSOJLSVHA UVAHSAUNM TI AA SAHAM ca AVA Td AVA IVMALSVM CUALVM M THA LVM M THSSHA THO LHHA IN IOdG A 0 IOd A TOAUVA TONYUVA VLAN YVA ULSO Sn IWDU 40 10 5 Differential Algebraic Equations As before we have r data sets 5 05 95 with ller and weights w Again weights can become zero in cases when the corresponding measurement value is missing if artificial data are needed or if p
188. The final m functions are the constraints g p for j 1 mr if they exist at all depending on the parameter vector p to be estimated Any other functions are not allowed to be declared The constants n m r and m are defined in the database of EASY FIT dce Design The last n of the n parameters to be estimated are considered as switching points if they have been declared to describe certain model changes Also ny the number of constant or variable break points must be defined a priori In addition to variables and functions a user may insert further real or integer constants in the function input file according to the guidelines of the language PCOMP Example The example was introduced in Section 2 5 of Chapter 2 Although am explicit solution is easily obtained we show here a possible implementation to illustrate the input of differential equations The system is given by two equations of the form j kiy 0 D Ya k y kayo y2 0 0 6 We assume that experimental data are available for both state functions y1 t and ya t and define the corresponding PCOMP code as follows G A a E E e C C Problem LKIN C G A E E E C VARIABLE ki k2 D y1 y2 t C FUNCTION yi t yi_t Kisel C FUNCTION y2_t y2_t ki y1 k2 y2 C FUNCTION y1 0 y1_0 D C FUNCTION y2 0 y2_0 0 C FUNCTION hi hi yl C FUNCTION h2 h2 y2 C END C 6 5 Differential Algebraic Equations Th
189. The line with number 1 denotes the left boundary of the integration area 29 Format Name Descr o Desert a6 4x 15 NCPB SC of area boundaries must be an even number since every area has a left and a right boundary OT a64xi5 NRES Number of constraints without bounds Eee Number of equality constraints a6 4x RT RX IX Formatted input of NRES rows each containing two real 2g20 4 15 numbers identifying the experimental time and spatial parameter values for which a constraint is to be sup plied and the corresponding line number The order is arbitrary but first the equality and subsequently the inequality constraints are to be defined The data are rounded to the nearest actual time value 20 a6 4x 215 NTIME Number of time points must be greater than 0 MPLOT Logarithmic scaling of x axis MPLOT 1 or not MPLOT 0 NFIT Number of fitting criteria 22 a6 4x 15 IFIT Formatted input of NFIT lines where each line contains the line number i e the discretization point where a fit criterion is defined The line with number 1 denotes the left boundary of the integration area 23 a6 4x i5 NPLOT Number of plot points to be computed by additional model function evaluations Plots are generated by in terpolation linear or polynomial depending on graph ics system NPLOT may be zero if no plot data are required 24 a6 4x 15 NOUT Output flag for PDEFIT NOUT 0 no generation of output files NOUT
190. Ti 1 Ti Or Lg E LE NE respectively k 1 r where r denotes the total number of measurement sets The fitting criterion may depend on the solution of the partial equation and its first and second derivative with respect to the space variable at the fitting point Fitting points are rounded to their nearest line when discretizing the system Basically each integration area is treated as an individual boundary value problem and is discretized separately by the method of lines The transition functions are treated in the same way as Dirichlet or Neumann boundary conditions respectively In order to achieve smooth fitting criteria and constraints we assume that all model functions depend continuously differentiable on the parameter vector p Moreover we assume that the discretized system of differential equations is uniquely solvable for all p with p lt p Pu A collection of 20 examples of partial differential equations that can be solved by the presented approach and comparative numerical results are found in Schittkowski 433 Example 2 23 HEAT_B To illustrate different integration areas and the application of transition conditions we again consider the heat equation but now formulated over two areas ul Dyul a Desst TT ur Dei 0 5 lt z lt 1 with diffusion coefficients D and D gt t gt 0 The initial heat distribution at t Le 2 104 ut p x 0 asin nrz 0 lt lt 05 2 105 u2 p z 0 asim zz
191. We have seen in the previous section that 32171 can be considered as an approximation of the covariance matrix o2 1 where x 1 l I 5 s p z Ju gt h p ti yi Zn i see 3 2 I Vf p V f p and p a least squares estimate for the true but unknown para meter p Assumptions are independent and normally distributed errors in the measurements with mean value 0 and variance o A more rigorous analysis based on the maximum likelihood function leads to the theorem of Cram r and Rao which states that the inverse of the Fisher information matrix is a lower bound for the covariance matrix of the parameter errors This matrix is approximately given by EEGEN 3 7 For a precise definition of this matrix and a proof see e g Goodwin and Payne 178 Since all induced matrix norms are greater than the spectral radius of a matrix we apply the L5 norm i e R a 1 IF ll Ama dE gt 3 8 Amin Lp 84 Exponential Test Function 8 7 6 5 4 h p t x 2 1 0 Hl zm 0 10 20 30 40 50 60 t Figure 3 1 Model Function and Data Amaz and Amin denote the largest and smallest eigenvalue of a matrix respectively Since small eigenvalues of Ip enforce large entries of the covariance matrix we try to reduce them by successive elimination of parameters corresponding to large eigenvector
192. a Gauss Newton and quasi Newton least squares method are retained see Schittkowski 429 for details In case of least squares data fitting with very many measurements the sum of squared functions is directly minimized by the SQP code NLPQLP 440 The total number of iterations might increase but the calculation time per iteration is decreased When minimizing a sum of absolute function values i e the L4 norm the problem is transformed into a smooth nonlinear programming problem by introducing 2 additional variables and inequality constraints The code is called NLPL1 L problems where the maximum of absolute residual values is to be minimized are solved by the code NLPINF 448 One additional variable and additional inequality con straints are introduced to transform the min max problem into a smooth nonlinear optimiza tion problem which is then solved by NLPQLP 440 In case of very many measurements the transformed problem is solved by the active set code NLPQLB 442 443 4 2 Steady State Systems The program MODFIT is executed to solve parameter estimation problems based on dy namical equations or steady state systems respectively To solve the corresponding systems of nonlinear equations they are treated as a general nonlinear programming problems and solved by the Fortran code NLPQLP see Schittkowski 427 440 449 Objective function is the sum of squares of the system parameters and the constraints are identical to th
193. a fifth order difference formula is applied The resulting ODE system is solved by an implicit method with integration accuracy 10 9 The least squares algorithm DFNLP of Schittkowski 429 with termination accuracy 10719 stops after 47 iterations at the optimal solution see Ta ble 2 13 for results Obviously we are able to identify the parameters also the parameter of the transition condition within the accuracy of the experimental data The final surface plot of the solution u p x t is shown in Figure 2 31 By the subsequent example we want to outline the possibility to define transition con ditions also for algebraic equations Another reason for presenting the example is to show that our approach allows to integrate also time independent systems of partial differential equations i e systems that do not contain any time derivatives at all 59 Heat Equation Q d bj y j b AN ji Figure 2 30 Continuous Transition Condition Heat Equation Figure 2 31 Non Continuous Transition Condition 60 p Po Pl p Du k 10 0 5 0 959 1 004 1 049 a 10 0 5 0 920 1 002 1 084 ag 10 0 5 0 943 1 004 1 064 Table 2 14 Exact Values Starting Values and Confidence Intervals Example 2 24 BEAM2 Two beams are clamped at two end point and linked at an in termediate point by some kind of joint Bending 1s modelled by a fourth order differential equation EL E w kw f z t 2
194. a for Ordinary Differential Equations The numerical analysis program MODFIT is executed for models based on systems of ordi nary differential equations Also in this case we need some integers that cannot be retrieved from the model function file or other data Number of Differential Equations Define number of ordinary differential equations The number of ODE s must coincide with the number of model functions for the right hand side and the initial conditions on the Fortran or PCOMP input file Number of Measurement Sets The number of measurement sets must coincide with the num ber of data sets as given in the input table for experimental data Number of Concentration Values The number of concentrations must coincide with the number of concentrations as given in the input table for experimental data If the value inserted is positive and the PCOMP input language is used then a concentration variable must be declared in the model function file If 1 is inserted it is supposed that the fitting criteria depend on an additional concentration variable and that one concentration value is assigned to each time value Shooting Index Shooting technique can be introduced in the following situation 1 There are as many measurement sets as differential equations 2 Fitting criteria are the system variables of the differential equation in exactly the same order 3 There are no additional constraints 4 There are no zero weights otherwise an
195. a s h eee AE A 2 22 Exypheot Model Punctione s ob awa Re oe ee sa ee ee ee ee 8 4 Zo Laplace Translorms ccosa c de omo or Q w dee dom ee ee 9 420 BESAN State Equations lt ss ss ede eee x afew se eee SE e W 11 2 5 Ordinary Differential Equations 25222 9344 a 4 14 221 Dlandatd Formulation o v nace NNN dh AN de chee W 14 2 5 2 Differential Algebraic Equations 00 15 250 BDwibhir Points es s cee ee bo ee SS owe A Ww x Rowe A 17 A METTE 26 208 LOGOS Method usu cosas mO ORM ee ae ee Cad 28 256 Boundary Value Problems uoo sum e 44 2 EO XR Ex 35 2f Variable Initial Times sec er ece ee RG See Re Q j eee Qu of 26 Partial Diferential Equations lt gt o lt ese 282424 84244 EO 254 8 4 43 20 1 Standard Formulation ee aia kom gad bee Bah A Q G 43 2 6 2 Partial Differential Algebraic Equations 45 2523 Flux Funeti ene Aus A s A a kc eee Rer Q W h pee wO QUA 48 2 6 4 Coupled Ordinary Differential Algebraic Equations 50 2 6 5 Integration Areas and Transition Conditions 56 2 6 6 Switching Points sce kee ee Ee ra sas NR eH RA 61 A l cue ace ome O3 ee ee ee SORS amp 65 217 Optimal Control Problems lt a cpe x RERUM Y x EX Vx eee Ee Su 71 Statistical Analysis and Experimental Design 79 3 1 Confidence Intervals 2222528 mm XR om m xe Xem ee DR ee HS EO 81 So Sjgnifeanee Levels sa oos at Nee 4a do e s k ep eee EA 83 3 3 Experimental Design 222 2 xoxo
196. able and that one concentration value is assigned to each time value Shooting Index Shooting technique can be introduced in the following situation 1 There are as many measurement sets as differential equations 2 Fitting criteria are the system variables of the differential equation in exactly the same order 3 There are no additional constraints 4 There are no zero weights otherwise an artificial weight of 1 0E 7 is inserted 5 There are no break points Integration is performed only from one shooting point to the next and then initialized with a shooting variable The differences of shooting variables and solution at right end of previ ous shooting interval lead to additional nonlinear equality constraints The shooting index determines the number and position of shooting points 0 no shooting at all 1 each measurement time is a shooting point 2 every second measurement time is shooting point 3 every third measurement time is shooting point etc Number of Algebraic Equations Define here the number of algebraic equations or algebraic variables respectively Note that the algebraic equations or algebraic variables respectively must follow the differential equations or differential variables respectively in the input file for model functions Number of Index 2 Variables If a higher index system is given define here the number of variables with index 2 The number is used to estimate the corresponding error and t
197. act the author under phone 49 0 921 553278 fax 49 0 921 35557 e mail klaus schittkowski Quni bayreuth de home page http www klaus schittkowski de 0 1 5 Dimensioning Parameters The numerical algorithms require dimensioning parameters for defining working arrays of suitable lengths They serve also as upper bounds for certain model parameters i e maxi mum number of variables to be estimated or maximum number of measurements Whereas the full version can be adapted to any size there are some restrictions for the demo version vi Number of parameters to be estimated 2 Number of equations 2 Number of constraints 5 Number of ODEs of discretized PDE 100 Number of time values 20 Number of measurement sets 2 Number of measurements 40 To find out the allowed maximum problem sizes of the full version one should investigate the corresponding include files MODFIT INC and PDEFTT INC from the utilities command of the menu bar New executable files can be linked subsequently if any of the bounds are changed and if object codes are available The meaning of the parameters used is completely described by initial comments vil Chapter 1 Introduction Parameter estimation plays an important role in natural science engineering and many other disciplines The key idea is to estimate unknown parameters pi pa of a mathema tical model that describes a real life situation by minimizing the distance of some k
198. active pulse transmission line simulating nerve axon Proceedings of the IRE Vol 50 2061 2070 Nagurka M L 1990 Fourier based optimal control of nonlinear dynamic systems Journal on Dynamical Systems Measurements and Control Vol 112 17 26 Nayfeh A 1972 Perturbation Analysis John Wiley New York Neittaanmaki P Tiba D 1994 Optimal Control of Nonlinear Parabolic Systems Marcel Dekker New York Basel Nelder J A Mead R 1965 A simplex method for function minimization The Computer Journal Vol 7 308 Nelson K A 1993 Using the glass transition approach for understanding chemical reaction rates in model food systems Ph D Thesis Minnesota University USA Nelson K A Labuza T P 1994 Water activity and food polymer science implications of state on Arrhenius and WLF models in predicting shelf life Journal of Food Engineering Vol 22 271 289 Nelson W 1981 Analysis of performance degradation data IEEE Transactions on Relia bility Vol 2 No 2 149 155 Newman P A Hou G J W Taylor A C 1996 Observations regarding use of advanced CFD analysis sensitivity analysis and design codes in MDO ICASE Report No 96 16 NASA Langley Research Center Hampton Virginia 23681 Newell R B Lee P L 1989 Applied Process Control A Case Study Prentice Hall Engle wood Cliffs New Jersey Nickel B 1995 Parametersch tzung basierend auf der Levenberg Marquardt Methode in Kombination m
199. add artificial weight factors to the observations at a predefined relatively dense grid specified in advance These weights are considered then as design parameters A particular advantage is that derivatives subject to weights are obtained without additional computational efforts Special emphasis is given to efficient computation of derivatives where first and second order partial derivatives of the model function of our dynamical system are approximated by forward differences The nonlinear constrained optimization problems are solved by the SQP code NLPQLP see Schittkowski 440 7 1 6 Optimization Tolerances A couple of tolerances and bounds are required to start the optimization algorithms for constrained nonlinear data fitting and experimental design The database provides suitable default values that can be used when a new problem is generated 6 Experimental Design Tolerances xj Computation Starting at Model Parameters Number of Parameters to Compute Covariance Matris and Confidence 0 4 Intervals Y Last Computed Parameter Set Number of Iterations 100 3 Weight Optimization SC Output Flag one line per iteration yes M no Termination Tolerance 1E 07 Qs Numerical Differentiation Tolerance To 4 Figure 7 4 Experimental Design Tolerances Number of Iterations A reasonable upper limit for the number of iterations is required Data Fitting One evaluation of the Jacobian matrix of the
200. aic equations Dimensioning parameter must be greater or equal to NCPL Number of measurement sets Dimensioning parameter must be greater or equal to NMEA Number of constraint functions in the parameter estimation prob lem Dimensioning parameter must be greater or equal to NRES When calling SYSFUN PAR contains the NPAR coefficients of the actual variables to be estimated PAR is not allowed to be altered within the subroutine When calling SYSFUN U contains the coefficients of the partial differential equations on the right hand side for the spatial dis cretization point X as described subsequently U is not allowed to be altered within the subroutine Function values to be evaluated in the case of IFLAG 3 for initial values of PDE s When calling SYSFUN UX contains the coefficients of the first derivatives of the solution of the partial differential equations for the spatial discretization point X and the transition conditions UX is not allowed to be altered within the subroutine When calling SYSFUN UXX contains the coefficients of the sec ond derivatives of the solution of the partial differential equations for the spatial discretization point X and the transition conditions UXX is not allowed to be altered within the subroutine Function values to be evaluated in the case of IFLAG 2 for right hand side of PDE s When calling SYSFUN V contains the coefficients of the coupled differential algebraic equations
201. ailable Thus we try to find a reasonable compromise which nevertheless leads to sufficiently stable procedure Differentiation of the objective function of 3 15 subject to qr 1 lt r nq gives Oo Ala z ace C p trace 7 e I p q i Oo trace 1 Feet x p q L p 07 3 16 trace 10 0 2 F p dia df 10 4 3 q P 3 trace 10 0 dE F p q 9 T sal F p Vee pq I p 4 There remains differentiation of the x n matrix F Exo DM Da p q Ee Vol P d 3 17 OH The mixed partial derivatives of the model function bin q t subject to p and q are approx imated by forward differences i l npik 1 l 2 1 h t RS h 141 r ryt h We Au Op p q tr ae h p gege q Erer tk h p q tk 3 18 h p q gp Erlr tr Ae h p T i q tk fork 1 Land i 1 np Here e IR and e JR are the i th and r th unit vectors respectively and e e are suitable perturbation tolerances e g chosen by e max l p e and e max 1 q e with a certain tolerance e gt 0 which must be selected very carefully 92 Equation 3 18 is written in a form to show that cancelation appears only once Since the evaluation of the objective function q i e of F p q V J p q requires also an approximation of first derivatives of the form a se gt por ees EE 3 19 only two additional evaluation of h i
202. ake and its Catchment Ecological Bulletins Vol 37 Stockholm Argentine M Coullet P 1997 Chaotic nucleation of metastable domains Physical Reviews E Vol 56 2359 2362 Ascher U M Mattheij R Russel R 1995 Numerical Solution of Boundary Value Problems SIAM Philadelphia Ascher U M Petzold L R 1998 Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations SIAM Philadelphia 12 13 14 15 16 17 18 19 20 21 22 Ascher U Ruuth S Wettin B 1995 Implicit explicit methods for time dependent partial differential equations SIAM Journal on Numerical Analysis Vol 32 797 823 Atkinson A C Bogacka B 2002 Compound and other optimum designs for systems of non linear differential equations arising in chemical kinetics Chemometrics and Intelligent Laboratory Systems Vol 61 17 33 Baake E Schloeder J P 1992 Modelling the fast fluorescence rate of photosynthesis Bul letin of Mathematical Biology Vol 54 999 1021 Baily J E Ollis D F 1986 Biochemical Engineering Fundamentals McGraw Hill New York Balsa Canto E Alonso A A Banga J R 2002 A novel efficient and reliable method for thermal process design and optimization Part I Theory Journal of Food Engineering Vol 52 227 234 Balsa Canto E Banga J R Alonso A A Vassiliadis V S 1998 Optimal control of dis tributed processes using restricted second
203. aluations Plots are generated by in terpolation linear or polynomial depending on graph ics system NPLOT may be zero if no plot data are required 19 a6 4x 15 NOUT Output flag for MODFIT NOUT 0 No output generated on files PRT TEX and RPL NOUT 1 Output generated on files PRT ETEX and RPI a6 4x 4i5 METHOD Choice of simulation or optimization s AA AA ISHT METHOD 0 Simulation NORM METHOD 1 Call of DENLP Schittkowski 429 NUMGRA METHOD 2 dummy ISHT is the shooting index to identify number and posi tion of shooting points If ISHT gt 0 only METHOD 0 1 or 2 are allowed ISHT 0 no shooting at all ISHT 1 shooting at every measurement time ISHT 2 shooting at every second time ISHT 3 shooting at every third time etc NORM determines the data fitting norm NORM 1 Li norm sum of absolute residuals NORM 2 La norm sum of squared residuals NORM 3 L norm maximum of absolute residuals NUMGRA must be set for gradient evaluation NUMGRA 1 analytical derivatives available NUMGRA 0 1 forward differences NUMGRA 2 two sided differences NUMGRA 3 5 point difference formula a6 4x i5 OPTP1 Parameter for chosen optimization algorithm METHOD 0 significance level 0 1 5 10 METHOD 1 maximum number of iterations a6 4x 15 OP TP2 Parameter for chosen optimization algorithm METHOD 1 maximum number of line search
204. alues and random errors based on a uniform distribution with relative deviation of 1 Subsequently confidence intervals subject to model parameters are computed as described in Section 3 1 see 3 6 The results are listed in Table 3 8 The maximum standard deviation is more than 800 i e it is practically impossible to estimate the model parameters based on the given design data The code MODFIT is executed with termination tolerance 1078 and e 10 for the ap proximation of partial derivatives The optimization routine NLPQLP of Schittkowski 440 terminates after 72 iterations after reducing the performance criterion from 2 7 x 10 to 6 9 10 Optimal design values are listed in Table 3 7 Corresponding state and control functions are shown in Figures 3 6 to 3 9 As for the initial design confidence intervals are computed for the design parameters see Table 3 8 Now all deviations are below 15 Table 3 7 Design Parameters before and after Experimental Design p initial final MV 1 0 0 24299 MV 0 3 1 13686 MV3 0 3 0 25613 Ja 0 75 0 80000 Used 0 5 0 67401 Dad 0 4 0 32820 Va 2 15 11 29776 Nal 0 106 0 68736 Na 0 106 0 18089 Nab 0 0876 0 30515 Talon 0 0319 0 78145 Na2eb 0 0319 0 17605 TA 0 0486 0 57685 Nabeb 0 0454 0 34219 98 10 F SZ XX 1 J d E 5 d Se oe ace E 0 10 20 30 40 50 60 70 80 Figure 3 6 State Functions ny t no t na t 14 12 PENNE 10 St E 6 4
205. ample is dry friction between two bodies see Example 2 10 below It must be expected that the direct integration of an ODE with discontinuities leads to numerical instabilities since very small stepsizes must be used to pass around a corner in the solution Typically the ODE formulation is extended by a vector valued switching function alo y p t t ap y p t t qu P y p 0 0 which must be given by a user a priori to specify the change of a sign in the model equations for example and which could also depend on parameters to be estimated Here we omit the additional concentration variable c to simplify the notation Then we proceed from the dynamical system in Fip y t sign alp y t yi 0 p Wa 2 44 Ym Fal 1 t sign q p y t gt Ym 0 2 y p Example 2 9 LKIN LA Consider again Example 2 8 We define q p y t t T and f 8 0 if sign q 2 1 _ Fi p Y t sign q p Y t E Etat if sign q ld gt Y1 0 Do gt 2 45 Falp y t sign q p y t kiyi Env y2 0 0 with p kao ka1 T The implicitly given switching times must be computed internally during the numerical integration of 2 44 As soon as a change of the sign of the switching function q p y t is observed a special root finding sub algorithm must be started to locate the switching time leading to substantial additional numerical efforts The integration is then restarted from the computed
206. an T P 2007 Modern Experimental Design Wiley Series in Probability and Statistics Saad M F Anderson R L Laws A Watanabe R M Kades W W Chen Y D I Sands R E Pei D Bergmann R N 1994 A comparison between the minimal model and the glucose clamp in the assessment of insulin sensitivity across the spectrum of glucose tolerance Diabetes Vol 43 1114 1121 Sakawa Y Shindo Y 1982 Optimal control of container cranes Automatica Vol 18 257 266 Sanz Serna J M Calvo M P 1994 Numerical Hamiltonian Processes Chapman and Hall London Saravacos G D Charm S E 1962 A study of the mechanism of fruit and vegetable dehy dration Food Technology 78 81 Schenk J L Staudinger G 1989 Computer model of pyrolysis for large coal particles in Proceedings of the International Conference of Coal Science Tokyo Schiesser W E 1991 The Numerical Method of Lines Academic Press New York London Schiesser W E 1994 Computational Mathematics in Engineering and Applied Science CRC Press Boca Raton Schiesser W E 1994 Method of lines solution of the Korteweg de Vries equation Computers in Mathematics and Applications Vol 28 No 10 12 147 154 Schiesser W E Silebi C A 1997 Computational Transport Phenomena Cambridge Uni versity Press Schittkowski K 1979 Numerical solution of a time optimal parabolic boundary value control problem Journal of Optimization Theory and Applications Vol 27
207. and must be smaller than the total number of discretized differential equations When inserting a zero value it is assumed that there is no band structure at all Break Points Similar to ordinary differential equations it is possible to define additional constant break points where the corresponding time values are to be defined in form of a separate table The integration is restarted with initial tolerances at these switching values The numerical values inserted must vary between zero and the maximum experimental time Copyright 2004 Ernst Hairer 44 Break Points Number of Break Points to be 3 A Optimized 4 Constant Break Points Figure 7 22 Break Points value Time values are ordered internally If the table contains no entries at all it is assumed that there are no constant break points with respect to the time variable On the other hand it is possible that the last n optimization variables to be estimated are used as break points in the code that defines the right hand side and the initial conditions In this case the number n must be inserted If n gt 0 and constant break points exist in the corresponding table then these values are ignored Consistency Parameters To compute consistent initial values the corresponding system of discretized algebraic equations is solved by the general purpose nonlinear programming method NLPQLP see Schittkowski 427 440 For executing NLPQLP a few parameters must be
208. and when estimating by J we get the approximate confidence intervals D GC di Di tes SV dii x 3 6 for the i th individual model parameter value p i 1 n In this case p is the i th coefficient of and d the i th diagonal element of f t see alo Gallant 161 or Donaldson and Schnabel 119 However 3 6 is valid only approximately depending on the quality of the linearization or the curvature of f p respectively Donaldson and Schnabel 119 present some exam ples where the confidence intervals are very poor Thus we have to be very careful when computing 3 6 without additional linearization checks Example 3 1 PARID15 30 60 120 We consider the model function pipa pat pit h p t e PY e n p SEH with three unknown parameters p p Do Dall to be estimated First we define a true parameter value p 0 1 1 100 7 and generate experimental data sets in the following way For 120 60 30 andl 15 we evaluate y h p ti where e is a normally distributed error with variance o2 0 01 at equidistant grid points t within the interval 0 60 1 1 Then we solve the corresponding data fitting problem 3 2 starting from po 0 05 2 120 with termination accuracy 107 Subsequently we compute the confidence intervals 3 6 as outlined above for the significance level a 1 see Soll 3 1 The corresponding lower and upper bounds ana the computed
209. ank G Corliss eds SIAM Philadelphia 315 330 Kahaner D Moler C Nash S 1989 Numerical Methods and Software Prentice Hall Englewood Cliffs Kalaba R Spingarn K 1982 Control Identification and Input Optimization Plenum Press New York London Kamke E 1969 Differentialgleichungen I Akademische Verlagsgesellschaft Geest und Por tig Kaps P Rentrop P 1979 Generalized Runge Kutta methods of order four with stepsize control for stiff ordinary differential equations Numerische Mathematik Vol 33 55 68 Karlsen K H Lie K A 1999 An unconditionally stable splitting scheme for a class of nonlinear parabolic equations IMA Journal of Numerical Analysis Vol 19 609 635 Kaps P Poon S W H Bui T D 1985 Rosenbrock methods for stiff ODE s A comparison of Richardson extrapolation and embedding techniques Computing Vol 34 No 1 17 40 Kelley C T 1999 Iterative Methods for Optimization SIAM Philadelphia Kim L Liebman M J Edgar T F 1990 Robust error in variables estimation using non linear programming techniques AIChE Journal Vol 36 985 996 Kim K V e al 1984 An efficient algorithm for computing derivatives and extremal prob lems English translation Ekonomika i matematicheskie metody Vol 20 No 2 309 318 Kletschkowski T Schomburg U Bertram A 2001 Viskoplastische Materialmodellierung am Beispiel des Dichtungswerkstoffs Polytetrafluorethylen Technische Mechanik Vol
210. ary differential equations with initial values 4 Filpy t c 0 yi p c n 2 31 Ym Po p us t c tal y9 p c s Without loss of generality we assume that as in many real life situations the initial time is zero The initial values of the differential equation system y p c y p c may depend on one or more of the system parameters to be estimated and on the concentration parameter c In this case we have to assume in addition that the observation times are strictly in creasing and get the objective functions r a lc Y Y Y ui NN ATE N 05 v5 2 32 2 2 f for the least squares norm T lt le gt Y Dd wih p u p ti cj ti 05 vis 2 33 1 i 1 1 for the L norm and k nibo gajagi kint A 1 A wy lhe lp yp NUN UD Vi 2 34 for the maximum norm The system of ordinary differential equations is to be solved numerically by explicit or implicit integration methods To be able to evaluate the gradient of the fitting criterion with respect to p for example to compute V f p wf V h p y p ti Geh ba Cj Vy p ti cj Vyh p y p ti Es ti 65 2 35 14 in case of 2 32 fori 1 l j 1 le and k 1 r we need the derivatives of the solution vector y p t c subject to p that is Vy p ti cj For evaluating Vy p t c we apply either outer approximations for example a forward difference or any similar formula we add sensitivity equations t
211. ast squares formulation J p s x Sou filv s t u p zj t uS p x t u i p Ta thutha NW TG j l 2 129 with suitable weights w The index k denotes the corresponding spatial integration interval containing zj In the second case the integral is replaced by Lin s 2x Y wil fa p s ti v p T ti uso T E ula T ti v p ti ti fat gt 1 2 130 where k denotes again the spatial integration area containing 7 Even in the third situation we are able to get a trivial least squares formulation J p s fp s T wp T ul 7 T uk v 7 T 0 T 2 131 if we knew that the algorithm we apply to solve the least squares problem is capable to handle situations where the length of the sum of squares is one Nonlinear state and control constraints may be added to the optimal control problem either in explicit form 2 50 for the discrete parameter vector p or in form of dynamical 72 inequality constraints 2 52 and 2 123 respectively where we have to add the dependency from the control variable say g p s t u p 7 t uti p 5 t ue p Tth Up t t 0 2 132 The discretized formulation leads to g p g p s t u p mi 15 6 pu b up 6 0 t 15 2 133 for j Me 1 My Here the index 7 denotes the corresponding integration area that contains the spatial parameter z rounded to its nearest line and t a suitable discrete time value where
212. ate the first and second partial derivatives of u p x t with respect to the spatial variable at a given point x several different alternatives have been implemented in PDEFIT a Difference Formulae First and second derivatives can be approximated by difference formulae see Schiesser 120 Difference formulae with 3 and 5 points for first derivatives are available that can be applied recursively to also get the second derivatives Alternatively a 5 point difference formula for second derivatives is implemented as well The difference for mulae are adapted at the boundary to accept given function and gradient values Moreover first derivatives can be approximated by simple forward and backward differences These formulae are recommended if there are steep fronts for example in case of transportation or fluid dynamics where symmetric procedures lead to numerical irregularities To apply one of these differences the flow direction must be known in advance They are particularly useful in a situation where an upwind formula is desirable but the right hand side of the PDE is not given in flux form see below Most of these difference formulae can be combined and applied individually to the spatial derivatives of the state variables under consideration b Upwind Formulae for Hyperbolic Equations In case of a scalar hyperbolic equation puesta 4 3 1 1 nm with a so called flux function f approximation by difference formulae might
213. ation values Eig system of ordinary differential equations with initial values concentration values and switching points o m system of differential algebraic equations up to index 3 with initial values concentration values and switching points system of one dimensional time dependent partial differential equations with initial and boundary values disjoint integration areas coupled ODE s flux functions and switching points same as for PDE but with additional algebraic partial differential equations and coupled DAE s ei E m z EE Anderson F Olsson B eds 1985 Lake Gaadsj on An acid forest lake and its catchment Ecological Bulletins Yol 37 Parameterstobe ne name lower bound starting value upper bound optimal PL Estimated pm 1 00E 03 1 0000E 00 1 3 pm 1 00E 05 1 0000E 00 1 00E 05 6 0377E 00 2 Update Dm 1 00E 05 1 0000E 02 1 00E 05 37099F 02 1 0 00E 00 1 0000E 00 j S L D Simulation wes rcow FORTRAN ES Data Fitting Nom Sum of Absolute Residual Values 7 Model Functions Sum of Squared Residuals Experimental Design Maximum of Residual values 8 m Record nl 4 I 1173 E ou p of 1398 Model name optional Figure 8 1 Main Form file with extension FUN or FOR contains the model functions either in the PCOMP input format or in form of a FORTRAN subroutine as specified by t
214. ble and we want to define a macro that computes the square of x we would write something like MACRO sqr Sqr x x Now it is possible to replace each occurrence of the term x x with an invocation of the macro that we have just defined for example f sqr 5 2 10 Figure 5 2 Piecewise Linear Interpolation SUM and PROD expressions Sums and products over predetermined index sets are for mulated by SUM and PROD expressions where the corresponding index and the index set must be specified for example in the form f 100 PROD p i a i i IN inda In the above example p i might be a variable vector defined by an index set and a i an array of constant data Control statements To control the execution of a program the conditional statements IF condition THEN statements ENDIF or IF condition THEN statements ELSE statements ENDIF 11 Figure 5 3 Piecewise Cubic Spline Interpolation can be inserted into a program Conditions are defined as in Fortran by the comparative operators EQ NE LE LT GE GT which can be combined using brackets and the logical operators AND OR and NOT The GOTO and the CONTINUE statements are further possibilities to control the exe cution of a program The syntax for these statements is GOTO label and label CONTINUE where label has to be a number between 0 and 9999 Since PCOMP produces labels during the generation
215. bog uornqr rsrp oui oouoptisoWd amp pnjs LSIN stuoje ourpor ur sj29jop tunquen opour 34018 SPIRYIY oseo oi Sni T suo1joo o Jo uorjen eAo AYLATOOYOY USISOP IO OO 10 opour uor2ogej Xpn3s LSIN sdo pue sq nq uoruo jo yystem AIG Xpn3s LSIN 9A1n5 YIM013 eprourgts YIM ppt 9 myseg uomoun feuorjez e Surg sjure1jsuoo UTM uoreurxoidde euoney soqo1d ado jostue jo Aj1suojur URW YY sjuourjreduroo poq ueumu OMY UL 199811 DALJIBOLPLH puno b39Dq T ON CH CH Ch Ch CH Ch CH Ch OOOO OO OOO CC Ch O Oh vu CO Ch CH oo noo Oh Ch OC Ch co eu CH Ch Ch Ch Ch OOO Ch OC O ei Gi OO c O CN CN CN CN CN CN CY Go GO Or Gi O r N 10 M OLT I 9 GS GE Kal VG GT IT IT 0 LT l 2 NN A Y hi cO CO CN si CO hi CN cO cO CN GO hi ONNA NNN N N Gi NN CY 0 d L c06d L CdL VIdL SId L TY TAL VIdL IdiL LOV ANIL Uaa NHL SAUNYAHL AWT ANAL ALVA TAS say dls ANITdS ONHLOOWS dX OMS ain INVINZSOH UO HOIN 4ASVa THY AJILOWTITWH Loa TAA SYLVH aw LIJ LVH ddV LVu NVINVH OVUL ON IWDU 10 ponurjuoo Sch 8cV Sch Sch Sch 8cV Sch Sch Sch Sch Sch Sch Sch Sch 8GP Sch Sch S i 8GP Sch Sch GGG 8GP Sch Sch Sch Sch Sch Sch fas swo oryerpenb moz uro qo 1d sorenbs 3seo poure ysuog Stu ejep orroeurouoSrr pue p erjuouodx zp Sun eyep peuorjeqq uorgeurrxoadde euorje y Sumu ejep perguouodx zq Surjjg eyep erguouodx q uro qoid uS
216. botern Dissertation Dept of Mathematics Technical University of Munich Heinzel G Woloszczak R Thomann P 1993 TOPFIT 2 0 Pharmacokinetic and Phar macodynamic Data Analysis System G Fischer Stuttgart Jena New York Henninger R J Maudlin P J Rightly M L 1997 Accuracy of differential sensitivities for one dimensional shock problems Report LA UR 97 2740 Los Alamos National Laboratory Los Alamos New Mexico 87545 Hilf K D 1996 Optimale Versuchsplanung zur dynamischen Roboterkalibrierung Fortschritt Berichte VDI Reihe 8 Nr 590 Hines A L Maddox R N 1985 Mass Transfer Prentice Hall Englewood Cliffs Hoch R 1995 Modellierung von Fliefwegen und Verweilzeiten in einem Einzugsgebiet unter station ren FlieBbedingungen Diploma Thesis Fakulty of Biology Chemistry and Geology University of Bayreuth Germany Hock W Schittkowski K 1981 Test Examples for Nonlinear Programming Codes Lecture Notes in Economics and Mathematical Systems Vol 187 Springer Berlin Hohmann A 1994 Multilevel Newton h p collocation ZIB Berlin Preprint SC 94 25 Hooker P F 1965 Benjamin Gompertz Journal of the Institute of Actuaries Vol 91 203 212 Horbelt W Timmer J Melzer W 1998 Estimating parameters in nonlinear differential equations with application to physiological data Report FDM University of Freiburg 15 226 227 228 229 230 231 232 233 234 235 236
217. braic equations SIAM Journal on Numerical Analysis Vol 27 1527 1534 Gear C W Osterby O 1984 Solving ordinary differential equations with discontinuities ACM Transactions on Mathematical Software Vol 10 23 44 Geisler J 1999 Dynamische Gebietszerlegung ftir Optimalsteuerungsprobleme auf vernetzten Gebieten unter Verwendung von Mehrgitterverfahren Diploma Thesis Dept of Mathematics University of Bayreuth Germany Gelmi C Perez Correa R Agosin E 2003 Modelling gibberella fujikuroi growth and GAS production in solid state fermentation Report Department of Chemical and Bioprocess En gineering Pontificia Universidad Catolica de Chile Casilla 306 Santiago 22 Chile Gibaldi M Perrier D 1982 Phamacokinetics Marcel Dekker New York Basel Gill P E Murray W 1978 Algorithms for the solution of the non linear least squares problem SIAM Journal on Numerical Analysis Vol 15 977 992 Gill P E Murray W Wright M H 1981 Practical Optimization Academic Press New York London Gill P E Murray W Saunders M Wright M H 1983 User s Guide for SQL NPSOL A Fortran package for nonlinear programming Report SOL 83 12 Dept of Operations Re search Standford University California Gill P E Petzold L Rosen J B Jay L O Park K 1997 Numerical optimal control of parabolic PDEs using DASOPT in L T Biegler T F Coleman A R Conn and F N Santosa eds Large Scale Optimization with A
218. c P 1998 Fixed point methods for computing the equilibrium composition of complex biochemical mixtures Biochemical Journal Vol 331 571 575 Kuzmic P 1999 General numerical treatment of competitive binding kinetics Application to thrombin dehydrothrombin hirudin Analytical Biochemistry Vol 267 17 23 Kuznetsov V A Puri R K 1999 Kinetic analysis of high affinity forms of interleukin 13 receptors Biophysical Journal Vol 77 154 172 Lafon F Osher S 1991 High order filtering methods for approximating hyperbolic systems of conservation laws Journal on Computational Physics Vol 96 110 142 Lambert J D 1991 Numerical Methods for Ordinary Differential Systems The Initial Value Problem John Wiley New York Lanczos C 1956 Applied Analysis Prentice Hall Englewood Cliffs Lagugne Labarthet F Bruneel J L Sourisseau C Huber M R Borger V Menzel H 2002 A microspectrometric study of the azobenzene chromophore orientation in a holo graphic diffraction grating inscribed on a p HEMA co MMA functionalized copolymer film Journal of Raman Spectroscopy Vol 32 665 675 Lang J 1993 KARDOS Kascade reaction diffusion one dimensional system Technical Report TR 93 9 ZIB Berlin Langtangen H P 1999 Computational Partial Differential Equations Lecture Notes in Computational Science and Engineering Vol 2 Springer Berlin Heidelberg Lapidus L Luus R 1967 Optimal Control of Engineering Pr
219. cance level then statistical error analysis data can be displayed on request They contain the following information Variance covariance matrix Correlation matrix EASY FIT Main Form l File Edit Start Report Data Delete Make Utilities Numerical report Analysis report Data and Function plot Model Data Exr Residual plot Surface plot Model E Optimization output Information onvection and Word report Project Number Demo Unit for SAM User Name Schittkowski Unit for A Figure 8 6 Report Command Estimated variance of residuals Confidence intervals for parameters Confidence intervals of estimated parameters are computed for the significance levels 1 5 or 10 respectively These data may help to distinguish between relevant and redundant parameters and to get an impression about the quality of the model and experimental data Moreover graphical plots show the fitting functions together with the experimental data given the individual residuals and in addition three dimensional plots 3D plots are gener ated only if concentration values are defined or in case of partial differential equations where the numerical solution functions of the system is displayed In both cases three dimensional surface plots are available Plots of model functions and experimental data are either generated by the internal plot program of EASY FIT de Pes s
220. cdd TAG d4sv Id4d AJLISNHG AVVA IVIIDHA dAH VOd UVH VOA NOD VOA qOOANVG X VO GINOD DAO HINOO DAD dOHdHNOD GLOHIMHO 9 Du Dono 9uou GIG D A E GN 0S gs 085 uou uou 9uou o9uou 9uou uou 0S m OS 030p r0 296 997 997 997 997 997 997 ptoloq od YIM prosdr o JO uorj2os1ojur o3 UIBO DIOU oou ejsrp umurxe Sj1ed xis LM Teor Xpn3s LSIN suerssnez popuo q A guo rjs OMT Apnys LSIN Suerssneg popuorq Apj3ms oA Xpn3s LSIN sueissnesy pojeredos o4 0A T suRISsnes JYSto 07 Ep JO SUNMA suomounj uerssnex jo uorjeurquioo reaut 03 Dou ooeds ueIsoyrey ut syurod jo uornqr3st Gq suorung uerssnex jo uorjeurquioo Ieu 03 po33r ooeds uersoj1e ut syurod jo uornqr3st Gq urmn35ods eurureg e jo sisApeuy opKgop urio JO uorjeururioj9p IJOWO OQ popou erqguouodxo oA J suo erqjuouodxo OMY jo wns uorjoung 9899 Por dx Y suo erguouodxo oA jo derroA Vep ur SIOLIO 93 18 Jopour erguouodxo w Sur poururiojopaopun suorjoung perurou amp qod suorjoung erurou amp qod suorjoung perurouAqod suorjoung re ur poururiojop oAo suorjoung euorgead poururrojopoAo suorjoung orrjourouoSLrm o durexo jsoT o durexo jsoT o durexo jsoT o durexo 389 o durexo 3sa o durexo 3sa uorjoung 9899 porn dx Y Surjjg eyep ergu uodxr Surjjg eyep ergu uodxr Surjjg eyep ergu uodxr Surjjg eyep ergu uodxrq AGO 1eo
221. ce x 0 2 Ges Ke 9 8 SIE with initial heat x L Vo Function v serves also as our fitting criterion Then the corresponding Fortran code is to be implemented as follows SUBROUTINE SYSFUN NPAR MAXPAR NPDE MAXPDE NCPL MAXCPL NMEA MAXMEA NRES MAXRES PAR U UO UX UXX UP V VO VP C CX FIT G DG X T IAREA LEFT RIGHT IFLAG FLUX FLUXX FLUXU FLUXUX FLUXPX IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION PAR MAXPAR U MAXPDE UO MAXPDE UX MAXPDE UXX MAXPDE UP MAXPDE V MAXCPL VO MAXCPL VP MAXCPL C MAXPDE CX MAXPDE FIT MAXMEA G MAXRES DG MAXRES MAXPAR FLUX MAXPDE FLUXX MAXPDE FLUXU MAXPDE MAXPDE FLUXUX MAXPDE MAXPDE FLUXPX MAXPDE LOGICAL LEFT RIGHT DOUBLE PRECISION K L eg SALSA SE De e S SS IF IFLAG EQ 0 RETURN C C SET PARAMETERS C L PAR 1 K PAR 2 PI 3 1415926535 C C BRANCH W R T IFLAG C GOTO 100 200 300 400 500 600 700 800 900 1000 1100 IFLAG RETURN C C EVALUATION OF FLUX FUNCTION C 100 CONTINUE RETURN c C RIGHT HAND SIDE OF PDE S 200 CONTINUE 41 Q Q Q Q Q Q Q C C C UP 1 UXX 1 RETURN INITIAL VALUES OF PDE S 300 CONTINUE UO 1 DSIN PI X L RIGHT HAND SIDE OF ODE S 400 CONTINUE VP 1 K PI L DEXP PI L 2 T RETURN INITIAL VALUES OF ODE S 500 CONTINUE VO 1 K L PI RETURN BOUNDARY FUNCTIONS 600 CONTINUE IF LEFT C 1 0 0 IF RIGHT C 1 0
222. coefficients The order by which the variables are eliminated can be considered as an indication about their relative significance the highest level reflects the highest priority We proceed from a given significance tolerance y gt 0 known experimental data and an optimal solution p of the corresponding least squares data fitting problem We try to satisfy 1 Amin TF Assuming a sufficiently accurate approximation of p the true parameter vector we hope to get sufficiently small variances Note that very small or zero eigenvalues lead to the conclusion that some parameters cannot be estimated at all by the underlying model and the available data or that there are combinations of highly correlated parameters see Caracotsis ans Stewart 75 To detect the significant parameters on the one hand and the redundant or dependent parameters on the other we apply the subsequent procedure see also Schneider Posten and Munack 451 or Majer 315 The idea is to successively eliminate parameters until 3 9 is satisfied The cycle is terminated in one of the following situations liz llo lt 3 9 1 The smallest eigenvalue of the Fisher information matrix is smaller than a threshhold value see 3 9 2 The parameter correlations are significantly reduced e g by 25 3 None of the above termination reasons are met and all parameters have been elimi nated 85 1 Algorithm 3 1 Let k 1 Jo 0 I VJG V E B
223. compiled automatically by EASY FIT 4 Pesis If the submitted default compiler interface is to be changed e g from the Watcom to the Salford or Lahey compiler the user has to set corresponding compiler name some path names and the compiler and linker execution commands An external editor can used to create or modify model functions either in the PCOMP or the Fortran language Note that EASY FIT Pesiz is delivered with two editors an internal GUI form EASY FIT and an external executable one with syntax highlight ing EDITOR EXE Both allow direct parse of PCOMP code or compilation and link of Fortran code To use the external editor the file EDITOR EXE must be part of the EASY FTT ModelPesis installation directory 0 1 4 Starting EASY FTTModelDesign It is recommended to start EASY FTT ModelDesign always from its shortcut in the program menu generated by the setup program or from a corresponding desktop icon to avoid con flicts with an existing Microsoft Office Access version The welcome window of EASY FIT 0 e Pesion is displayed and the main form of the database is opened If the main form cannot be opened correctly please check the language settings Non unicode languages like Chinese Arabic or other settings cause some problems If the database reacts too slow for example when starting a data fitting code or when displaying a report delete a certain subset of problems you do not need V The file README TXT c
224. ctory with object codes for the Watcom Salford Lahey Compaq Absoft Microsoft and Intel For tran compilers containing underlying optimization algo rithms and ODE PDE solvers only complete version PROBLEMS Directory for test example files with extensions lt gt FUN and lt gt FOR 0 1 3 System Setup Download the file EASYFIT EXE and start the installation by clicking on this file In case of a local network administrator rights are required If there exists an older version of EASY FIT VodelDesign it is recommended to save first all problems of interest to a temporary directory After successful installation the saved problems can be imported again EASY FIT WVodelDesi n comes with the royalty free runtime version of Microsoft Office Access 2007 When starting EASY FIT 9 the first time after a successful setup a couple of directory strings are inserted automatically into an internal table They can be adapted to a special situation depending on the environment given Alterations can be made by the Utilities command in the menu bar E g the favorite text editor may be defined to be used for input and modification of model functions In more detail the following configuration information is available ill System Configuration Data x System Directory C Easyfit E asy_fit mdb Editor EpTOREXE z Graphics System fEASYFIT z Fortran Compiler INTEL z Default Export Import Path ICumgwp o User Name
225. d in form of general equality or inequality constraints g p 0 J 1 Me 2 27 gp gt 0 j m 1 m It is assumed that all functions are continuously differentiable subject to p The above formulation 2 27 includes also the possibility to define dynamical inequality restrictions that are constraints depending on the state variable z p t c at known time and concentration values Thus constraints of the form g p g p Z p tij Ck o tijs Ck 2 28 for m lt j m are permitted at predetermined time and concentration values that must coincide with some of the given independent measurement data If constraints are to be defined independently from given measurement data it is recommended to insert dummy experimental values with zero weights at the desired time and concentration points t and Ck respectively A more extensive discussion and an example is found in the subsequent section Example 2 4 RECLIG19 To illustrate the data fit of a steady state system consider the following example that is similar to a receptor ligand binding study with one receptor and two ligands see Schittkowski 432 z 1 p122 p223 ps 0 zo 1 p121 Pa 0 j 2 29 23 1 poz t State variables are zi z2 and za and the parameters to be fitted are p po and pa pa 100 is fixed to avoid an overdetermined system and t is the independent model variable There is no concentration variabl
226. d in the variable section of the input file as outlined above 2 Model functions defining the right hand side of the partial differential equations F p T fiu Uy Ure U x t rera Fs b a fests Ur Ura U x t are defined next one set for each integration area i 1 ma Each function may depend on z t v U Uz Ure p and optionally also on the flux functions and their derivatives In this case the corresponding identifiers for fluxes and their derivatives as specified in the variable section must be used in the right hand side 10 10 The corresponding initial values at time 0 are set next ui p z i 1 Ma They depend on z and p and are given for each integration area separately Next the n coupled differential equations must be defined in the order given by the series of coupling points i e functions G p U Us Ugo v t J 1 Ne where the state variable u is evaluated at a given discretization line together with its first and second spatial derivatives Then initial values of the coupled ordinary equations at time 0 are defined vi pg 1 ty The Subsequently n Dirichlet transition and boundary conditions are set in the order given by the area data first left then right boundary functions ci p u v t c p u v t where function values of u at the left or right end point of an integration area are inserted Neumann transition and boundary c
227. d in the database In case of an initial analysis a simulation is performed with respect to the given set of parameters as provided by the user or the data fitting iteration cycle is started from these parameters Otherwise a restart is performed i e the simulation or optimization run started with parameters from the database that are calculated by a previous run In this case the user is asked whether these values should replace the existing parameter values or not Note that the numerical results are sent to the database only if the displayed window is not closed before the numerical code terminated completely and an information message is visible If a PCOMP input file for declaring model functions is chosen then the statements can be parsed directly from the corresponding form If on the other hand an external editor is preferred then the input cannot be parsed directly and the last line of the Start menu offers the corresponding command to execute the parser EASY FIT Main Form l File Edit Report Data Delete Make Utilities Data fitting Restart of Data Fitting Code Information Diffusion of water through soil convection and Project Number Demo Unit for 4 User Name Schittkowski Unit for AM Figure 8 5 Start Command 8 4 Report Command The report command serves to produce reports and function and data plot A text report is generated directly from the database and is displayed
228. d om the IR Then some real valued functions f defined on IR n 1 m are called a sequence of elementary functions for f m gt n if there exists an indez set J with J C 1 i 1 Ji nj for each function fi i n gt 1 m such that any function value of f for a given vector p pi Pn can be evaluated according to the following program For 1 n 1 m let pi fi px k Ji 5 1 Let f p Pm The proposed way of evaluating function values is implemented in any compiler or in terpreter of a higher programming language if we omit possible code optimization consid erations In computer science terminology we would say that a postfix expression is built in the form of a stack which is then evaluated recursively Thus the elementary functions can be obtained very easily and the corresponding technique is found in any introductory computer science textbook Note that for every function f p there exists at least one trivial sequence of elementary functions by m n 1 and f p f p For practical use however we assume that the functions f are basic machine operations intrinsic or external functions where the relative evaluation effort is limited by a constant independently of n Under this condition suitable bounds for the work ratio can be proved The algorithm can be implemented efficiently by using stack operations which reduce the storage requirements as far as possible i e we do not
229. d too large errors in the measurements In addition the constraint functions may depend on the solution of the dynamical system at predetermined time and concentration values i e g p 3 p y p bis Chr z p tijs Ce tij Ck 2 52 for any j m lt j lt m where y p t c and z p t c denote the solution of a differential algebraic equation 2 36 depending on the parameter vector p to be estimated and certain time and concentration values t and c respectively Note that dynamical constraints should be defined only in form of inequalities Equality conditions can be handled as algebraic equations and are part of the dynamical model The predetermined time and if available at all concentration values must coincide with some of the given experimental data If constraints are to be defined independently from given measurement data it is recommended to insert dummy experimental values with zero weights at the desired time and concentration points and ck respectively 7 me 1 ee or Population Dynamics 2 200 2 000 7 o ign 3 1 800 But h p y p t t 1 600 1 400 1 200 1 000 0 0 00 0 1 0 15 0 2 0 25 0 3 035 0 4 Figure 2 14 Function and Data Plot
230. data These test examples can be used to check the accuracy of discretization formulae or ODE solvers Moreover we show some references in the column headed by ref from where further details can be retrieved Either the data fitting problem is described in detail or at least the mathematical background of the model is outlined In case of an empty entry the model is provided by private communication and not published somewhere else or a related reference is unknown to the author To summarize we offer test problems for the following model classes explicit model functions 245 Laplace transforms 10 steady state equations a 41 ordinary differential equations 575 differential algebraic equations 62 partial differential equations 325 partial differential algebraic equations 44 sum 1 300 10 1 Explicit Model Functions We proceed from r measurement sets of the form m Eeler ala yal mn with l time values le concentration values and ller corresponding measured exper imental data Moreover we assume that weights UE are given However weights can become zero in cases when the corresponding measurement value is missing if artificial data are needed or if plots are to be generated for functions for which experimental data do not exist Thus the subsequent table contains the actual number lt 1 of terms taken into account in the final least squares formulation Usually we proceed from the L or Euclidean norm to
231. data fitting func tion with respect to the variables to be estimated Experimental Design One evaluation of the covariance matrix of the experimental design performance measure It is recommended to start with a sufficiently low number say 50 If the resulting answer is not as precise as wanted a restart can be performed where the last computed iterates are inserted as starting values for another optimization run Output Flag There is the possibility to control the desired output shown on the screen 1 only the iteration number and the actual residual value displayed 0 no output 1 only final convergence analysis 2 one line per iteration 3 more detailed output information per iteration 4 in addition also line search data are displayed Output level 2 is recommended The original output of the selected least squares algorithm is directed to a file with extension HIS depending on the print flag chosen Only the residual values are displayed on screen in a DOS window executing the numerical code However one has to be a bit familiar with the underlying mathematical theory to understand the data in detail For the meaning of the parameters displayed it is necessary to read the corresponding user guides Gradient Evaluation EASY FIT 9 comes with two options to define model func tions Fortran code to be compiled or the PCOMP modeling language In the latter case gradients are evaluated automatically and are inserted whene
232. de fines the general model structure 28 1 system of partial differential equations 2 system of partial differential algebraic equations Lr m ms eem eme PROJECT Plot output first line of information block e g project number TEST Plot output second line of information block e g mea surement characterization DATE Plot Plot output third line of information block e g date Plot output third line of information block e g date line of information block e g date ESL E Z AXIS Name for z axis spatial variable ls falo Y AXIS Name for y axis value 9 falo T AXIS Name for t axis time 10 a6 4x 2i5 NPAR Number of parameters to be optimized must be at least one NBPV Number of variable break points i e the last NBPV variables are treated as break points where integration is restarted NPDE Number of PDE s must be at least one NPAE Number of algebraic equations 13 a6 4xj5 NCPLO NCPLA Numbers of coupled ordinary differential and algebraic equations 14 a6 4x 15 ICPLO Formatted input of NCPLO lines where each line con tains the line number ie the discretization point where the ODE is coupled to the PDE The line with number 1 denotes the left boundary of the integration area 15 a6 4x 15 ICPLA Formatted input of NCPLA lines where each line con tains the line number ie the discretization point where the algebraic equation is coupled to the PDE
233. defined only for the implicit integration routine Break Points It is possible that the right hand side of a system of ordinary differential equations is non continuous with respect to integration time e g if non continuous input functions exist or if the model changes at certain time values In case of constant break or switching points respectively the corresponding time values are to be defined in form of a separate table where the integration is restarted with initial tolerances The numerical values inserted must vary between zero and the maximum experimental time value Time values are ordered internally If the table contains no entries at all it is assumed that there are no constant break points with respect to the time variable On the other hand it is possible that the last n optimization variables to be estimated are used as break points in the code that defines the right hand side and the initial conditions In this case the number n must be inserted If n gt 0 and constant break points exist in the corresponding table then these values are ignored Constraints Restrictions are allowed for models based on ordinary differential equations and can be formulated in form of equality and inequality constraints with respect to the parameters to be optimized and the solution of the dynamical system at some of the given experimental time and concentration values Where the total number of constraints can be retrieved from the subseq
234. dence intervals are computed for the design parameters K and K In the next step we consider the feed controls at 19 grid points as additional design parame ters and 20 constraints are added to prevent that C s t falls below zero The perturbation 93 tolerance for gradient approximations by forward differences is set to e 0 01 NLPQLP needs 18 iterations to reduce the performance criterion from 1 3 10 to 0 009 under termi nation accuracy 10 5 Optimal design parameters are the initial concentrations CH 38 9 and C9 14 3 and the optimal feed curve is shown in Figure 3 3 In Figures 3 4 and 3 5 the corresponding state functions C s t and Cx t are plotted After getting the optimal design parameters the confidence intervals are computed in the same way as for the starting values Standard deviations are reduced from 239 8 to 0 093 for K and from 0 0096 to 0 0022 for K Moreover the correlation coefficient is reduced from 0 99 to 0 19 Figure 3 3 Control Function Fin t 80 Ur 7 60r 7 590r 7 40 K 1 30r 7 207 1 10r 1 00 10 20 30 do 50 60 70 80 90 Figure 3 4 State Function C s t 94 PNW gt ODA OO 0 10 20 30 40 50 60 70 80 Figure 3 5 State Function C x t 95 Example 3 5 URETHAN1 2 A practically relevant example is studied by Bauer et al 29 the reaction of urethane The corresponding DAE describing the reaction of phenyliso cyanate ni butanol n
235. ditions between the different areas may be defined in addition They are allowed at most at transition points and have the form w pz t c pi E50 5 u p x t ch p u B SE u t gt 2 98 ui AD aj t s cP p b EZ a it uL pu Ek v t gt l ur p zu co p u p x t ui p xF t v t with 0 lt t lt t i 1 m 1 A transition condition of the i th area either in Dirichlet or Neumann form depends on the time variable the parameters to be estimated and the solution of the neighboring area Again the user may omit any of these functions if a transition condition does not exist at a given x value More complex implicit boundary and transition condition can be defined in form of coupled algebraic equations Since the starting time is assumed to be zero initial conditions must have the form u p x 0 ui p T T leia 2 99 and are defined for all z GE z 2 1 Ma If initial values for algebraic variables are not consistent i e do not satisfy the e equations of 2 96 the given values can be used as starting values for solving the corresponding system of nonlinear equations by Newton s method If the partial differential equations are to be coupled to ordinary differential algebraic equations we proceed from an additional DAE system of the form j Gy p u p j t ud p z t ud p 2 t v t 2 100 for j 1 Tae and 0 Gy p u p e t ui p 23 4
236. dth of Jacobian 4 Figure 7 11 ODE Parameters Note that the first two codes use dense output i e the integration is performed over the whole interval given by first and last time value and the intermediate solution values are interpolated In these cases gradients are obtained by external numerical differentiation If constant or variable break points are given the integration is restarted at these time values with all initial tolerances supplied Gradients are obtained by external numerical differentiation The explicit algorithm is capable to evaluate derivatives of the solution of the ODE internally i e by analytical differentiation of the Runge Kutta scheme If constant or variable break points are defined then the usage is prevented for some internal reasons Final Absolute Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the absolute global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible It is recommended to start with a relatively large accuracy for 22 example 1 0E 6 together with a low number of iterations and to increase the accuracy when approaching a solution by restarts Final Relative Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the relative global error is to be inserted
237. dual kp at EA km 1 30 0 000063 3 167 3 403 10 804 7 041 2 8 0 000375 3 382 3 320 13 667 6 709 3 6 0 000125 2 951 3 763 12 210 7 008 Table 2 9 Performance Results and Computed Solution Case 8 of Table 2 9 contains achieved performance results and computed parameters The best residual is obtained for case 1 But the model is incorrect as in case 2 For case 3 we get the most appropriate results 40 Pharmacodynamic Model z1 t c sx gt A 0 5 EE 1 gt 04 0 6 e Figure 2 21 State Variable x t c over Time t and Concentration c Pharmacodynamic Model lt aaa SE EE Figure 2 22 State Variable x2 t c over Time t and Concentration c 41 Pharmacodynamic Model Figure 2 23 State Variable x3 t c over Time t and Concentration c 42 2 6 Partial Differential Equations 2 6 1 Standard Formulation Now we proceed from r data sets A s 2 69 where l time values and lr corresponding measurement values are defined To simplify the analysis we omit the additional ind
238. e nonlinear system of equations given The algorithm proceeds from a successive quadratic approximation of the Lagrangian function and linearization of constraints To get a search direction a quadratic programming problem must be solved in each iteration step A subsequent line search stabilizes the algorithm Also the starting values required to initialize an optimization cycle must be predeter mined by the user in a suitable way They may depend on the parameters of the outer opti mization problem The system of nonlinear equations must be solved for each experimental time and concentration value Moreover the gradients of the model function h p z p t t are calculated analytically by the implicit function theorem In this case a system of linear equations must be solved for each time value by numerically stable Householder transforma tions 4 3 Laplace Back Transformation MODFIT is also executed to solve parameter estimation problems where the model func tions are defined in the Laplace space In this case constraints are not allowed Model functions and gradients are either declared in form of Fortran code or through the automatic differentiation features of the PCOMP language If an analytical back transformation is not available we have to apply a numerical quadra ture formula see Bellman et al 33 for details In our case we use the quadrature formula of Stehfest 490 Proceeding from a given Laplace transform H p s c
239. e Laplace space we consider Yi s k D E s ki 5 ka The functions are the Laplace transforms of two simple linear differential equations If measurements are given for both functions we define a model function file in the following way C Problem LKIN_L C Los ua ax ni mU dii Eie oma si a C VARIABLE k1 k2 D s C FUNCTION Y1 Y1 D s k1 FUNCTION Y2 Y2 ki D s k1 s k2 END 6 3 Systems of Steady State Equations In this case our system variables must be declared in the following order 1 2 5 The first n names identify the n independent parameters to be estimated pi Pn The subsequent m identifiers define state variables of the system of nonlinear equations Zl Zm If a so called concentration variable c exists a corresponding variable name must be added next The last name identifies the independent time variable t for which measurements are available Any other variables are not allowed to be declared Model functions defining the algebraic equations constraints and fitting criteria are defined as follows 1 5 The first m functions are the right hand sides of the steady state equations s p z t c ele The subsequent m functions define starting values for solving the system of equations which may depend on the parameters to be estimated on the time variable and eventually also on the concentration variable z p
240. e Model Parameters xi Number of Algebraic Functions Y E Conetrdint onstraints Number of Measurement Sets 1 a lt Number of Concentrations 0 4 Qi Iteration Bound for Equation Solver 50 8 a E Line Search lteration Bound Output Flag for Equation Solver 0 Final Accuracy for Equation Solver 1E 10 Figure 7 8 Parameters for Steady State Equations Number of System Functions The number of functions of the algebraic system must coincide with the number of functions to be declared in the model function file The fitting criteria are not counted Number of Measurement Sets The number of measurement sets must coincide with the num ber of data sets as given in the input table for experimental data Note that a data set corresponds to two input columns in the table for values and weights Number of Concentration Values The number of concentrations must coincide with the number of concentrations as given in the input table for experimental data If the value 16 inserted is positive and the PCOMP input language is used then a concentration variable must be declared in the model function file If 1 is inserted it is supposed that the fitting criteria depend on an additional concentration variable and that one concentration value is assigned to each time value Maximum Number of Iterations for Solving Nonlinear Equations For solving the internal systems of equations by NLPQLP the maximum number
241. e and the fitting criterion is h p z t pa z3 Experimental data are simulated at 13 time values 1 5 10 50 100 500 1 000 000 and subject to p 0 01 pz 0 0005 and pg 1 An error of 5 is added to the measurement values For our numerical tests we use the starting values A ps 2 pa and z9 t for solving the system of nonlinear equations pj 0 1 po 0 0001 and p4 2 are the 12 Steady State System 0 6 0 4 h p z t 0 3 0 2 0 1 1 10 100 1 000 10000 100 000 1 000 000 t Figure 2 4 Model Function and Data Plot starting values for the least squares algorithm DFNLP of Schittkowski 429 executed with a termination tolerance of 109 After 11 iterations we reach the parameters p 0 0109 p2 0 000554 and pa 0 985 The final residual is 0 00020 Model function values and simulated measurement data are plotted in Figure 2 4 13 2 5 Ordinary Differential Equations 2 5 1 Standard Formulation As before we proceed from r data sets of the form Ge udo E DEE bol 2 30 where l time values le concentration values and liler corresponding measurement values are defined The vector valued model function h p y p t c td halo y p t c t 0 help y p t tel depends on the concentration parameter c and in addition on the solution y p t c of a system of m ordin
242. e declare suitable lower and upper bounds O lt p lt 1 for the parameters to be estimated i 1 10 Experimental data are simulated subject to a parameter vector p JR and 83 equidistant time values 0 005 0 01 0 015 0 4 An error of 5 is added to the 26 o Bi p pi Di 1 0 05 0 05 0 2 0 20 0 1717 2 0 05 0 30 0 2 0 18 0 1715 3 0 05 0 60 0 2 0 17 0 1713 4 0 05 1 00 0 2 0 15 0 1711 5 0 60 0 05 0 2 0 12 0 0760 6 0 60 0 30 0 1 0 08 0 0765 7 0 60 0 60 0 1 0 05 0 0763 8 0 60 1 00 0 1 0 03 0 0761 9 1 20 0 05 0 1 0 01 0 0048 10 1 20 0 30 0 1 0 01 0 0049 Table 2 5 Constants and Staring Optimal and Computed Parameter Values for Population Dynamics computed simulation data The remaining constants a and Bi the starting value p the optimal parameter p and the computed parameter p are summarized in Table 2 5 Numerical results are obtained by an explicit Runge Kutta method with absolute and relative termination accuracy 1077 The least squares solver DFNLP is started with a termination tolerance of 1079 After 18 iterations the termination conditions are satisfied at p with a residual value of 0 00093 scaled by sum of squares of measurement values Initially the equality constraint is violated but is satisfied on termination subject to an error of 0 26 10 15 A function and data plot is shown in Figure 2 14 Obviously we are unable to recompute the known exact solution vector p One possible reason is the we generate
243. e following order of PCOMP variables is required 1 6 The first variable names are identifiers for n parameters to be estimated pi Pn The subsequent ma names identify the differential variables yi Yma The subsequent 7m names identify the algebraic variables 21 Zma If a concentration variable exists another identifier must be added next to represent C The last variable name defines the independent time variable t for which measurements are available Any other variables are not allowed to be declared Similarly we have rules for the sequence by which the model functions are defined 1 2 The first ma functions define the differential equations F p y z t c Fm p y 2 t c The subsequent m functions are the right hand sides of the algebraic equations i e functions G1 p 9 2 0 6 Ga p yY z t c Subsequently ma functions define initial values for the differential equations which may depend on the parameters to be estimated and the concentration variable y p c 0 zip s Pe Then m functions define initial values for the algebraic equations which may de pend on the parameters to be estimated and the concentration variable in c 0 Zma D C Next r fitting functions hi p y z t c hr p y z t c must be defined depending on p y z t and c where y and z are the differential and algebraic state variables of the D
244. e not permitted in this case Test problems defined by their Laplace transforms are listed in Table B 2 Table B 2 Laplace Transforms name n l background ref data CONCS 2 7 Test problem only concentration values S5 DIFFUS_L 1 99 Linear diffusion with constant parameters S1 ELEC NET 6 30 Electrical net S5 LKIN_L 3 26 Simple linear compartment model Laplace formula E tion LKIN_L3 2 78 Simple linear compartment model three initial doses S5 PLASTER1 7 7 Pharmaceutic transdermal diffusion plaster 552 194 E PLASTER2 4 7 Pharmaceutic transdermal diffusion plaster 552 194 E PLASTER3 5 12 Plaster diffusion 552 194 E PLASTERA 2 12 Plaster diffusion 552 194 E TRAY 3 26 Marking tray hits E 14 10 3 Steady State Equations Again it is supposed that r measurement sets of the form De Je Lesla B lar are given with l time values le concentration values and liler corresponding measured E data Moreover we have weights wr Weights can become zero in cases when the cor responding measurement value is missing if artificial data are needed or if plots are to be generated for state variables for which E data do not exist Thus the subsequent table contains the actual number lt l of terms taken into account in the final least squares formulation Together with an arbitrary fitting criterion A p z t c we get the parameter estimation problem min 2e Dti E wi he p z p ti c5 ti c UU yi 9 P 0 E L
245. e problem within EASY FITVodelDesign A particular advantage is the possibility to define a filter on the actual database and to select certain subsets of problems with predetermined properties Model name Arbitrary string to identify the type of a model Information Long information string up to 80 characters Project number Could contain project or any related information User name Is set to the user name in the database when generating a new problem Measurement set Identification of measurement data Date Inserted automatically when generating a new problem Unit for X values Any string for identifying the X or time axis in plots Unit for Y values Any string for identifying the Y or measurement axis in plots In a separate window also names for individual measurement sets are defined Unit for Z values Any string for identifying the Z or concentration axis in 3D plots Memo Text field for input of additional documentation text Moreover fitting criteria can be identified by some strings used in the generated reports and plots 7 1 3 Parameters to be Estimated The names for the optimization parameters to be estimated their upper and lower bounds and their starting values must be defined in form of table records where the initial value must not be smaller than the lower bound or greater than an upper bound The variable name is arbitrary and used only for generating readable output Note that the order in
246. e the evaluation of the Jacobian of the right hand side of the discretized system of ordinary differential equations with respect to the system parameters since they have to apply Newtons method to solve certain systems of nonlinear equations The Jacobian is evaluated either numerically by the automatic differentiation features of PCOMP or must be provided by the user in form of Fortran statements In all cases it is possible that the Jacobian possesses a band struc ture depending on the discretization scheme EASY FIT Pess allows to define the bandwidth that is passed to the numerical integration routines to solve systems of internal nonlinear equations more efficiently The bandwidth is the maximum number of non zero 37 entries below and above a diagonal entry of the Jacobian and must be smaller than the total number of discretized differential equations When inserting a zero value it is assumed that there is no band structure at all The bandwidth can be defined only for the implicit integration routine CET M 4 Number of Break Points to be 3 a Optimized L 4 Constant Break Points Figure 7 19 Break Points Break Points Similar to ordinary differential equations it is possible to define additional constant break points where the corresponding time values are to be defined in form of a separate table The integration is restarted with initial tolerances at these switching values The numerical values inserted must vary
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248. each switching point DFNLP is started with termination tolerance 107 from zero control values where an im plicit solver is applied to integrate the ODE with final accuracy 1079 Table 2 17 contains numbers of iterations nj and final residual values r for n 10 20 40 Figures 2 39 to 2 41 show the substrate distribution x t over the time axis Figure 2 42 the corresponding one for xa t and Figure 2 43 the exact control function s t ake akigexp ks1t and the approximated ones If we apply non continuous control functions as in the previous example it is necessary to restart the integration at each switching point to avoid the generation of extremely 73 n Nit T 10 63 0 43 107 20 92 0 34 1079 30 108 0 47 1071 Table 2 17 Performance Results Nonlinear Kinetics of a 3 Blocker Figure 2 39 State Variable x t for n 10 Nonlinear Kinetics of a P Blocker 18 06 18 04 18 02 18 X1 t 17 98 17 96 17 94 17 92 0 0 2 0 4 0 6 0 8 1 Figure 2 40 State Variable x t for n 20 74 Nonlinear Kinetics of a P Blocker 18 17 995 17 99 Su 17 985 17 98 17 975 0 0 2 0 4 0 6 0 8 1 t Figure 2 41 State Variable x t for n 40 Nonlinear Kinetics of a 3 Blocker 10 9 8 7 6 X t 5 4 3 2 1 0 0 0 2 0 4 0 6 0 8 1 Figure 2
249. easurement Sets d A Number of Concentrations 0 Constraints pie Figure 7 6 Parameters for Explicit Model Functions Constraints Restrictions are allowed for explicit model functions and can be formulated in form of equality and inequality constraints with respect to the parameters to be optimized and some of the given experimental time and concentration values Where the total number of constraints can be retrieved from the subsequent table the number of equality constraints must be supplied Equality restrictions must be defined first 13 in the input file for model functions values for which constraints are to be defined Order Name Time c x value Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or concentration parameter The number of lines in the table must coincide with the number of constraint functions defined on the model Serial order number of constraints Equality constraints must be defined first Arbitrary name for the constraint to be printed in reports Corresponding time value at which a constraint is to be evaluated Note that the time values are rounded to the nearest experimental time value to avoid a complete re integration of the system In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does
250. ed in the subsequent sections But explicit model functions can reflect solutions of dynamical systems that are analytically solvable as shown by the subsequent example 5 h p t t 0 0625 0 0714 0 0823 0 1 0 125 0 167 0 25 0 5 1 0 2 0 4 0 Vi 0 0246 0 0235 0 0323 0 0342 0 0456 0 0627 0 0844 0 16 0 1735 0 1947 0 1957 hip ts vil 2 5230 4 6890 2 7982 5 4681 3 9113 2 6394 8 5962 1 3765 8 6748 3 6381 8 0491 10 4 10 1074 1078 1078 1078 1078 error 0 0 20 0 0 9 16 0 8 6 4 2 10 2 8 6 5 0 0 2 0 0 Table 2 1 Experimental Data and Final Residuals 0 2 Rational Approximation 0 18 0 16 0 14 0 12 0 1 0 08 0 06 0 04 0 02 0 5 1 Figure 2 1 Function and Data Plot 1 5 Hh 2 5 3 5 Linear Compartmental Model Figure 2 2 Function and Data Plot Example 2 2 LKIN_X3 The nezt test case consists of an explicit solution of a linear ordinary differential equation hi p t c cexp pit ha p t c m E exp pat exp pit The concentration parameter represents now the initial dose of an experiment and is set to Cy 50 ca 100 and c3 150 We assume that there are measurements for both fitting criteria at time values 1 2 3 4 5 10 15 20 25 30 40 50 60 Experimental data are simulated by inserting
251. efine the final accuracy by which the subproblem is solved In case of exact derivatives a value between 1 0E 8 and 1 0E 12 is recommended Constraints Restrictions are allowed for models based on differential algebraic equations and can be formulated in form of equality and inequality constraints with respect to the parameters to be optimized and the solution of the dynamical system at some of the given experimental time and concentration values Where the total number of constraints can be retrieved from the subsequent table the number of equality constraints must be supplied Equality restrictions must be defined first in the input file for model functions The table allows to define time and concentration values for which solution values of the underlying dynamical system can be inserted 31 Order Serial order number of constraints Equality constraints must be defined first Name Arbitrary name for the constraint to be printed in reports Time Corresponding time value at which a constraint is to be evaluated Note that the time values are rounded to the nearest experimental time value to avoid a rest of the integration of the system In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does not exist c x value Corresponding concentration value at which a constraint is to be evaluated Note that concentration values must be rounded to the
252. eis peuroq r amp a c 3 a F amp 8or v sux yy ACO reoumuoN Ioyeay 10j A Y JO O13u00 xoeqpooq 811998U1 orjse o snoost A 811998U1 orjse o snoost A SIIOAIOSOI YUL Surjo 1oju uro qoid oS reqostp opp Surpuo q ureorjs pmbrT UOTJVII I9PIO YI U YIM AUR Sur SUOTINGIISIP Up SULIO oSesop pr os Jo oseopo1 ojerpouruiq UOTNGIIsIP ueurojeg sur10 oS sop PIJOS JO oseo o1 ojerpouruiq UOTINGIISIP IInqI AA SULIO oS esop pr os JO oseopo1 o3erpouruiq punosbyo0q So oo C EN EN CHE Q O OC Q O 00 Go Occ Oc Occo oo So SO OOOO CHE C ma C Q co coc O O O C O co Oo o vi o lt lt cO lt CN aO E r rw oc C W n moi CN om HOO AA sh GO IG 08 OV 0 8cI 06 F 6I 00 ST vr 9I 99 GL OVE 6 9 OT 6 TSP D l OV TT 8I SVG 86 86 NN 00 GO CV C 00 E e CQ CQ 70 P COO CN CN CN P lt lt xi aao CN cO CO 10 C N g A4MVLAN ANVLOML HOVISOME ANOOM L XHOML NOLL XING LL Han NAGgD L Idd TAL OHOLLL SATA L CSOSNVUL TSASNVUL ULSOLAOL ANNO TOL NYIHHL AdO Lt INOOJdNAL CNO TAAL INO LTH L CAHMNVL SIC MNVE CIIANVL ANVL 8SIT JVL LSIG qdVL 9SKT AVL IWDU 39 SS SS gS gS gS SS SS SS SS 0S gS SS SS 030p 9cT TEG GOP Oct Oct c9 OVE 9v 661 661 9cT vez ves uogxue dooz pue uojxue doj3 qo jo ymor oinj no jseoK s 1oxecp snonurjuoo Surje Ds
253. el parameters So far we proceeded from a given experimental design and try to fit some model para meters However the initial design might not be the best one and the question is how to improve or even optimize it Possible design parameters are time dependent input feeds ini tial concentrations or temperatures The goal is to construct a suitable performance criterion depending on design parameters additional constraints as far as necessary and to solve the resulting nonlinear optimization problem Since the confidence intervals mentioned above are mainly determined by the diagonal elements of the covariance matrix a possible objec tive function is the trace of this matrix see Section 3 3 Special emphasis is given to the efficient computation of derivatives where first and second order derivatives of the model function of our dynamical system are all approximated by forward differences To show that this approach is nevertheless a quite stable and efficient procedure if carefully implemented two examples are included The first one is a microbial growth model see Banga et al 22 which consists of a small system of only three differential equations two model parameters and design parameters in form of initial concentrations and an input feed However one of the model parameters is extremely difficult to estimate and the authors decided to apply a stochastic search method The other example is intensively investigated by Bauer et al 29
254. ependent model variable c called concentration in the previous sections In its most simple form a system of time dependent one dimensional partial differential equations is given by u F p u Uz Uie A 2 70 The expanded form is Qui Ec F U Ug Ugg Y 8t 1 p 2 71 Oun PF U Ug Ura L ER Dr if we consider the individual coefficients of F and u i e if F p U Ux ge t Fi p U Ux Uze og Fnp D U Uz Wan T SE and u u1 us respectively We denote the solution of 2 70 by u p x t since it depends on the time value t the space value x and the actual parameter value p Also initial and boundary conditions may depend on the parameter vector to be esti mated Since the starting time is assumed to be zero initial conditions have the form u p x 0 uo p z 2 72 and are defined for all x rj xg For both end points x and zz we allow Dirichlet or Neumann boundary values np u p t u p zg t u pt 2 73 us p 21 t wu p t p ma t F p t for 0 lt t T where T is the final integration time for example the last experimental time value t The availability of all boundary functions is of course not required Their particular choice depends on the structure of the PDE model for example whether second partial derivatives exist in the right hand side or not 43 To indicate that the fitting criteria h p t depend also on the sol
255. erformed Proceeding from the assumption that the model is sufficiently linear in a neighborhood of an optimal solution vector and that all experimental data are Gaussian and independent the following statistical data are evaluated e Variance covariance matrix of the problem data e Correlation matrix of the problem data e Estimated variance of residuals e Confidence intervals for the individual parameters subject to the significance levels 196 596 or 1096 respectively The number of model parameters for which the covariance matrix and the confidence intervals are computed is typically the number of all given parameters But there are situations where these parameters are divided into model and design parameters for example and where the statistical data are to be evaluated only for a certain subset Thus the number of these variables can be restricted to those for which a statistical analysis is computed The given value corresponds to the first ones found in the table and are called the model variables Data Fitting The mathematical background of the applied algorithms is described in Schitt kowski 429 The basic idea is to introduce additional variables and equality constraints and to solve the resulting constrained nonlinear programming problem by the sequential quadratic programming algorithm NLPQLP of Schittkowski 427 440 It can be shown that 4 Simulation Tolerances I D x Computation Starting at Model Parameter
256. erische Mathematik Vol 68 129 142 Schittkowski K 1997 Parameter estimation in one dimensional time dependent partial differential equations Optimization Methods and Software Vol 7 No 3 4 165 210 Schittkowski K 1998 Parameter estimation in a mathematical model for substrate diffusion in a metabolically active cutaneous tissue Progress in Optimization I 183 204 Proceedings of the Optimization Day Perth Australia June 29 30 Schittkowski K 1998 Parameter estimation and model verification in systems of partial differential equations applied to transdermal drug delivery Report Dept of Mathematics University of Bayreuth Germany Schittkowski K 1999 PDEFIT A FORTRAN code for parameter estimation in partial differential equations Optimization Methods and Software Vol 10 539 582 Schittkowski K 2002 EASY FIT A software system for data fitting in dynamic systems Structural and Multidisciplinary Optimization Vol 23 No 2 153 169 Schittkowski K 2002 Numerical Data Fitting in Dynamical Systems A Practical Intro duction with Applications and Software Kluwer Academic Publishers Dordrecht Boston London Schittkowski K 2004 PCOMP A modeling language for nonlinear programs with auto matic differentiation in Modeling Languages in Mathematical Optimization J Kallrath ed Kluwer Norwell MA 349 367 Schittkowski K 2006 NLPQLP A Fortran implementation of a sequential quadratic pro gra
257. es 59 n case A case B 7 0 752 347 16 0 752 591 03 11 0 752 634 19 0 752 634 20 21 0 752 633 46 0 752 633 47 Table 2 12 Neumann Boundary Condition Versus Coupled Algebraic Equation 2 6 5 Integration Areas and Transition Conditions Now we extend the model structure to take different integration intervals into account Possible reasons are the diffusion of a substrate through different media for example where we want to describe the transition from one area to the next by special conditions Since these transition conditions may become non continuous we need a more general formulation and have to adapt the discretization procedure The general model is defined by a system of ng one dimensional partial differential equa tions and n algebraic equations in one or more spatial intervals see also Schittkowski 433 These intervals are given by the outer boundary values x and p that define the total inte gration interval with respect to the space variable x and optionally some additional internal transition points 27 in Thus we get a sequence of m 1 boundary and transition points Ly e XX ed su 2 95 For each integration interval we define a system of partial differential equations of the form L D y UN UA D QU Vm PU VUES E D En An fin Eet 2 96 0 mE Fi p f p ul ie T t f p 505 46 Du ut ous t H where x JR is the spatial variable with z lt x lt x fori 1 ma
258. esc Tiie pe mn f g p gt 0 gt I Me 1 My pi lt p lt p We assume that fitting criteria hy p z t c k 1 r state variable z p t c and con straints g p j 1 my are continuously differentiable functions subject to p The state variable z p t c IR is implicitly defined by the solution z of a system si p Sei 0 Sala 0 of nonlinear equations All steady state test problems are listed in Table B 3 Since none of them possesses additional constraints the corresponding figures m and m are omitted 15 penurquoo a ev a Ierd q Ierd q Ierd q evi as Ier a IS srt IS eS ar GS eS soe Gs Gs Gs IS sog a a GS a IS a a a 020p LOV LOV LOV LOV LOV LOV LOV LOV EZS ves T6I 96 96 96 GOP Sal cca Spue3t omg pue s103doooa1 OMY HL opour umriqrmbo sse ajomdumb jo am 3uouroove dst T 9A M9 uorjemjeg spuesi om 103dooo1 9uo Lu 9A MO quouroove dst T purest ouo 109d9991 ouo YIM oA1no quourooe dst Kpn3s Surputq puesi 103doooqT pues ouo pue 10 da001 ouo ou iquiour 103d9991 uo punodurioo Heg JAMI uorjeinjeg Apnys Surputq puesi 103dooo1T Kurouo1jse JO uro qo1d res nq pezrpeuriou ojejs Kpeojs 103989 xu polis MO SNONULJUO oje3s Apeojs ur 19378 1LLSO 9931s MOJ UoS XO YIM ouet jour jo uorjeprxo TOL umriqipmbo 1ourer 93 19urtp I9urouo v umriqmmbo 1ourrp rourouo
259. esis Dept of Electrical En gineering and Computer Sciences University of California at Berkeley Scott M R Watts H A 1976 Solution methods for stiff differential equations in Numerical Methods for Differential Systems L Lapidus W E Schiesser eds Academic Press New York London 197 227 Seber G A F 1977 Linear Regression Analysis John Wiley New York 30 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 Seber G A F 1984 Multivariate Observations John Wiley New York Seber G A F Wild C J 1989 Nonlinear Regression John Wiley New York Seelig F F 1981 Unrestricted harmonic balance II Application to stiff ODE s in enzyme catalysis Journal of Mathematical Biology Vol 12 187 198 Seifert P 1990 A realization of the method of lines used for chemical problems Colloquia Mathematica Societatis Janos Bolyai Numerical Methods Vol 59 363 373 Sellers P J Dickinson R E Randall D A Betts A K Hall F G Berry J A Collatz G J Denning A S Mooney H A Nobre C A Sato N Field C B Henderson Sellers A 1997 Modeling the exchanges of energy water and carbon between continents and the atmosphere Science Vol 275 502 509 Seredynski F 1973 Prediction of plate cooling during rolling mill operation Journal of the Iron and Steel Institute Vol 211 197 203 Severo A M Diaz Romanach M L B Rodriguez L M P Ginart J
260. et to zero a suitable tolerance is internally computed depending on the order of the differentiation formula and an estimated accuracy by which the data fitting function is evaluated Termination Tolerance The tolerance is used to stop the algorithm as soon as some internal termination conditions are satisfied and should not be smaller than the accuracy by which derivatives are computed 1 0E 6 Final Residual Estimate A rough estimate for the expected final residual value can be in serted in case of least squares norms 0 01 If the code breaks down after very few iterations a larger step could lead to shorter steps and a more stabilized procedure Confidence Level If the significance level for determining confidence intervals is positive a statistical analysis is performed Similar to a simulation run he following statistical data are evaluated where the termination tolerance is used for determining the rank of the correlation matrix e Variance covariance matrix of the problem data 8 e Correlation matrix of the problem data e Estimated variance of residuals e Confidence intervals for the individual parameters subject to the significance levels 1 5 or 10 respectively In addition significance levels are evaluated to identify the significance of parameters and to help to decide upon questions like which parameters can be identified which parameters should be kept fixed whether additional experimental data mu
261. ew extensions see Dobmann Liepelt Schittkowski and Trassl 116 for details In particular the declaration and exe cutable statements must satisfy the usual Fortran input format i e must start at column 7 or subsequently A statement line is read in until column 72 Comments beginning with C at the first column may be included in a program text wherever needed Statements may be continued on subsequent lines by including a continuation mark in the 6th column Either capital or small letters are allowed for identifiers of the user and key words of the language The length of an identifier has to be smaller than 20 tokens In contrast to Fortran however most variables are declared implicitly by their assignment statements Variables and functions must be declared separately only if they are used for automatic differentiation PCOMP possesses special constructs to identify program blocks PARAMETER Declaration of constant integer parameters to be used throughout the program par ticularly for dimensioning index sets SET OF INDICES Definition of index sets that can be used to declare data variables and functions or to define sum or prod statements INDEX Definition of an index variable which can be used in a FUNCTION program block REAL CONSTANT Definition of real data either without index or with one or two dimensional index An index may be a variable or a constant number within an index set Also arithmetic expressions ma
262. expressions An analytical expression is as in Fortran any allowed combina tion of constant data identifiers elementary or intrinsic arithmetic operations and the special SUM and PROD statements Elementary operations are xr Note that PCOMP handles integer expressions in exponents in the same way as real ex pressions and one should avoid non positive arguments Integer constants are treated as usual artithmetic operations Allowed intrinsic functions are ABS SIN COS TAN ASIN ACOS ATAN SINH COSH TANH ASINH ACOSH ATANH EXP LOG LOG10 SQRT Alternatively the corresponding double precision Fortran names possessing an initial D can be used as well Brackets are allowed to combine groups of operations Possible expressions are 5 DEXP z i or LOG 1 SQRT ci f1 2 INDEX Variables In PCOMP it is possible to define indices separately to avoid unnecessary differentiation of integer variables They have to be defined in the program block INDEX INDEX 13 1 It is allowed to manipulate the index by statements of the form 1 2 4 3 a 1 a i 2 i 2 0 SUM a m i m IN ind i gG Hh Fh h hh H P I H In this case a must be declared in form of in integer array However the following assignment statements are not allowed if b is a real array i b 3 i 1 0 i 4 2 f i 3 0 Interpolation functions PCOMP admits the interpolation of user defined data using either a piecewise consta
263. external programs evaluating these data The format is described in detail among the initial comments of the Fortran code of PDEFIT Numerical results are stored on a file with extension RES and are read by EASY FTT ModelDesign as soon as numerical execution is terminated In case of a simulation run with a positive significance level statistical data are written to a file with extension STA in abbreviated form The code distributed together with EASY FIT Pess allows the input of model functions on a user provided file with name lt MODEL gt FUN in a directory specified in the first line of PDEFIT DAT In this case the input format is the PCOMP language must be used as outlined in the previous chapters A specific advantage is the automatic evaluation of derivatives There is no further compilation or link necessary and PDEFIT can be started immediately as soon as both input files are created An input file named PDEFIT DAT must contain the parameter estimation data some problem information and optimization data in formatted form The first 6 columns may contain an arbitrary string to identify the corresponding input row if allowed by the format 1 a80 FILE Name of output files generated by PDEFIT The string may begin with a path name but must not contain an extension Suitable extensions are selected by PDEFIT 2 a6 4x 15 MODEL Problem name passed to subroutine SYSFUN for iden tifying data fitting models The subsequent integer
264. f p u pat t uz p 27 t t uy p 2 171 t Gap pue Tio tur p 2 4 0 t for 1 lt k lt np In this case the derivative values at the boundary are replaced by the given ones before evaluating the second order spatial derivative approximations Ordinary differential equations are added to the discretized system without any further modification Since arbitrary coupling points are allowed they are rounded to the nearest line of the discretized system In the same way fitting criteria can be defined at arbitrary values of the spatial variable When defining the transition function it is important to have the underlying flux di rection in mind If for example the flux is in the direction of the spatial variable and we want to define a continuous transition at xf then we have to formulate the corresponding transition function in the form uj p z t ui p x t in order to guarantee that the boundary values at z are spread over the interval For the same reasons outlined in the previous sections it is possible that the right hand side of a PDE becomes non continuous with respect to integration time Thus it is possible to supply an optional number time values where the integration of the DAE is restarted with initial tolerances The integration in the proceeding interval is stopped at the time value given minus a relative error in the order of the machine precision Break or switching points are either constant or
265. f lines they have to be rounded to the nearest neighboring line Also initial values v p 0 vo p 2 93 may depend again on the parameters to be estimated A solution of the coupled system depends on the spatial variable z the time variable t the parameter vector p and is therefore written in the form v p t and u p x t To indicate that also the fitting criteria hi p t depend on the solution of the differential equation at the corresponding fitting point and its first and second spatial derivatives we use the notation hi p t hi p u p Tk t u D Tk t zz D Tk t OLD t t 2 94 see also 2 74 k denotes the index of a measurement set Also fitting points z are rounded to their nearest line when discretizing the system Coupled ordinary differential equations can be used to define a fitting criterion for ex ample if the flux into or out of a system is investigated Another reason is that they allow to replace Dirichlet or Neumann boundary conditions by differential equations see the sub sequent example Example 2 21 SALINE The model describes the diffusion of drug in a saline solution through membrane see Spoelstra and van Wyk 89 There are two differential state vari ables c and z with corresponding system equations G Fate z cs qc and Z qc 52 p Po p 0 1 1 0 0 103 D b 0 3 1 0 0 184 q 1 0 0 5 1 420 Co 5 0 1 0 5 063 Table 2 11 Exact Starting and Computed Pa
266. for plant growth Gartenbauwis senschaft Vol 56 No 3 99 106 Ricker W 1954 Stock and recruitment Journal of Fishery Research Board Canada Vol 211 559 663 Robertson H H 1966 The solution of a set of reaction rate equations in Numerical Analysis J Walsh ed Academic Press London New York 178 182 Roberson R E Schwertassek R 1988 Dynamics of Multibody Systems Springer Berlin Rominger K L Albert H J 1985 Radioimmunological determination of Fenoterol Part I Theoretical fundamentals Arzneimittel Forschung Drug Research Vol 35 No 1 415 420 Roos Y H 1995 Phase Transition in Foods Academic Press San Diego Rosenau P Str der A C Stirbet A D Strasser R J 1999 Recent advances in modelling the photosynthesis Report IWR University of Heidelberg Rosenbrock H H 1969 An automatic method for finding the greatest and least value of a function Computer Journal Vol 3 175 183 Ross G J S 1990 Nonlinear Estimation Springer Berlin Rudolph P E Herrend rfer G 1995 Optimal experimental design and accuracy of para meter estimation for nonlinear regression models used in long term selection BNiomedical Journal Vol 37 183 190 or 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 Runge C 1895 Uber die numerische Aufl sung totaler Differetialgleichungen Mathema tische Annalen Vol 46 167 178 Ry
267. forced by adding artificial interpolation data The spline functions generated are twice differentiable with the exception of the fourth break point At this point there exists only the first derivative and PCOMP generates the right hand side differential quotient for the second derivative We need at least four pairs of data to construct a spline interpolation as outlined above To given an example we assume that we want to interpolate the nonlinear function f t given by the discrete values f t y from Table 5 1 using the different techniques mentioned above Interpolation functions are defined by a program block starting with the keyword CONINT for piecewise constant functions LININT for piecewise linear functions or SPLINE for piecewise cubic splines followed by the name of the function The numerical values of the break points and the function values are given on the subsequent lines us ing any standard format starting at column 7 or later Using piecewise constant approximations we get for our example 8 i ti Yi 1 0 0 0 00 2 1 0 4 91 3 2 0 4 43 4 3 0 3 57 5 4 0 2 80 6 5 0 2 19 7 6 0 1 73 8 7 0 1 39 9 8 0 1 16 10 9 0 1 04 11 10 0 1 00 Table 5 1 Interpolation data CONINT F 0 0 0 00 1 0 4 91 2 0 4 43 3 0 3 57 4 0 2 80 5 0 2 19 6 0 1 73 7 0 1 39 8 0 1 16 9 0 1 04 10 0 1 00 Within a function definition block the interpolation functions are treated as intrinsic Fortran functions that is they have to
268. formulate a parameter estimation problem of the form 2 16 min ia Xa Xa wbu5tue wE g p 0 H 3 2 lisame gt pe R f g p 20 j m 1 m where we assume that fitting criteria hy p t c k 1 r and constraints g p j 1 m are continuously differentiable functions subject to p The model function h p t c hy p t c h p t c does not depend on the solution of an additional dynamical sys tem and can be evaluated directly from a given parameter vector p that is to be estimated at given time and concentration values t and c All explicit test problems are listed in Table B 1 ponurquos a gS IN uou g a Domp Daag D g S S 9c 1 cst tot 6 Str 8zp cv oer qos eyep ut 39s eyep pag qos eyep pug qos eyep 4ST S schter Re P Xpn3s LSIN 3po q ou j oruosedj je3sourouo 91e13s Ape 1 0061 93 064T Sre i9Ao snsuoo Sf quoted Jo SUTUSPILH wo qoid uorjeredos 10yesdTeyey GQ00 0 sde yya uorjenbo s 193 Mmg Jo uomnos yor dx Xpn3s LSIN pueurop uoS xo eorur uoorsq Sjureijsuoo AYLI enbo pue uorjnjos ye uerqooef JUATOYep YURI YYLM s330g Jo uro qo d 3so T Sjure1jsuoo Kyrpenbo pue 3183s ye uetqooe f 3uorogop JULI YIM sssog Jo uro qod 3s9T Aypqegnuopr uoN amp pu3s LSIN Suropour uorjezrjouseur yrarnpuoo1odug Jopour re ur Surputq oseqo urdoad3y sjuouromseour re norguo odeys suo ortgudsv sju
269. formulation 2 1 2 2 or 2 3 by fs p wy hip tej yi 2 14 where s runs from 1 to ller in any order Moreover we assume that there are suitable weight factors wk gt 0 given by the user that are to reflect the individual influence of a measurement on the whole experiment Zero weights can be defined if for example there are several concentration values c1 cj but measurements are not available for each time value t f The basic idea is to minimize the distance between the model function at certain time and concentration points and the corresponding measurement values This distance is denoted as the residual of the problem In the ideal case the residuals are zero indicating a perfect fit of the model function by the measurements In addition we allow any nonlinear restrictions on the parameters to be estimated in form of general equality or inequality constraints g p l a Me gt glo gt ES 0 j 0 J Me 1l Mp It must be assumed that all constraint functions are continuously differentiable with respect to p To summarize the resulting least squares problem is of the form min Xa Xia Ej wh help tac V5 gj P 0 J 1 Me gt peli Wa 2 16 gp 2 0 4 Jg qna de eos Mes see 2 1 Alternatively we get the corresponding L formulation by minimizing T lt le k 2 gt gt wy lhe p ti cj isl k 1 i 1 j 1 see 2 2 or the L formulation k k
270. h ETEMA e User Name Schitkowski Plot Configuration Path Edit GNUPLOT Commands r Overlays for Standard Plots o 4l Grid Lines for Standard 3D Plots T Initial XAngle for 3D Plots bp Horizontal Increment of 3D Plots bp Initial Z Angle for 3D Plots fis Vertical Increment of 3D Plots fio Figure 8 17 Configuration Form 21 Generated Time values x Distribution of Time Values ais na Tine C Exponential Number of Time Values 10 Shift fi Generate Time Values 3 Figure 8 18 Generation of Time Values Randomly Generated Errors x Error Distribution Standard Deviation Figure 8 19 Random Errors 22 Chapter 9 External Usage of Numerical Codes It is possible to execute the numerical codes MODFIT and PDEFIT also outside of the interactive user interface of EASY FIT 9 A reason could be to solve of a large number of parameter estimation problems controlled by a separate code of the user In this case a data input file is requested that contains all information to start the numerical algorithm The format of the file is documented in this chapter in detail Test cases illustrate the usage of the codes and their numerical performance 9 1 MODFIT Parameter Estimation in Explicit Model Functions Steady State Systems Ordinary Dif ferential Equations and Differential Algebraic Sys tems Basically MODFIT is a double precision Fortran subrou
271. he structure of the PDAE model for example whether second partial derivatives 60 exist in the right hand side or not Moreover arbitrary implicit boundary conditions can be formulated in form of coupled algebraic equations However we must treat initial and boundary conditions with more care We have to guarantee that at least the boundary and transition conditions satisfy the algebraic equations 0 Falp u p L t Uz p L t o u Bit s 0 Falp up ER t us p ER t ues D R t V R t If initial conditions for discretized algebraic equations are violated that is if equation 0 F p u p x 0 uz p x 0 Up T 0 v p a 0 is inconsistent after inserting Dirichlet or Neumann boundary values and corresponding approximations for spatial derivatives the corresponding system of nonlinear equations is solved internally proceeding from initial values given Each set of E data is assigned a spatial variable value x rr xg k 1 r where r denotes the total number of measurement sets Some or all of the z values may coincide if different measurement sets are available at the same local position Since partial differential equations are discretized by the method of lines the fitting points z are rounded to the nearest line The resulting parameter estimation problems is min Song Pom wi hi p u p Tk ti Ua Un Testa Uas Dees ti UC ti ti We yy ne IR Op U 5 1 m gj
272. he table allows to define time and spatial values for which solution values of the underlying dynamical system can be inserted 46 Order Serial order number of constraints Equality constraints must be defined first Name Arbitrary name for the constraint to be printed in reports Time Corresponding time value at which a constraint is to be evaluated Note that the time values are rounded to the nearest experimental time value to avoid a reset of the integration of the system In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does not exist c x value Corresponding spatial parameter value at which a constraint is to be evaluated The values are rounded to the nearest line number Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or spatial parameter The number of lines in the table must coincide with the number of constraint functions defined on the model function input file either Fortran or PCOMP AT Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 24 Constraints 48 Chapter 8 Menu Commands The menu commands of EASY FIT Model Design to define or alter data and functions to start an optimization run or to get
273. he user Possible reasons for exporting data are the execution of the numerical codes outside of the database or the possibility to copy problem data to another system Note that these text files can be imported again by as outlined The File command also allows to define a filter for selecting a subset of parameter esti mation problems from the database The search mask contains the following items EASY FIT Main Form Edit Start Report Data Delete Make Utilities x rn Model Design Find Ac Save as Import imental Data Export Diffusion of water through soil convection and Apply Filter Exit FTO SCUIN Br Demo Unit for SA User Name Schittkowski Unit for Y V Figure 8 2 File Command Problem Identifier Model Name Information Project Number User Name Measurement Set Date The first six strings may contain a for determining a group of problems If a date is defined then all problems are selected with date less than or equal to the given one The corresponding database query assumes that at least some of the input fields contain non empty strings Finally the filtered problems can be sorted subject to the same input fields either in ascending or descending order ES select Problem to be Generated or Replaced Figure 8 3 Save As Command 8 2 Edit Command All data that define the dynamical model can be defined and changed by moving to the corresponding input field directly
274. he wants to accelerate the function evaluation The order in which functions must be coded and the organization of corresponding subroutine arguments is described within initial comments of the frame in serted automatically when generating a new problem There is only one subroutine for the evaluation of all model functions and gradients depending on an input flag If the subrou tine is called with zero flag then additional statements or subroutine calls can be inserted to prepare data before starting the optimization cycle In case of explicit fitting functions and dynamical systems of equations analytical gradient evaluation may be omitted Then numerical approximations based of forward differences are applied In case of ordinary dif ferential equations gradients must be programmed only if an ODE solver is to be executed that requires the computation of derivatives either by an implicit method or by of internal evaluation of derivatives 7 1 5 Numerical Analysis EASY FIT dc Pesis allows to perform a simulation with respect to given set of parameter values to start an iterative data fitting run or to compute an optimal experimental design Simulation A numerical analysis either at the set of given parameter values or the parameter values obtained from a previous data fitting run is performed Moreover plot data are generated If the significance level for determining confidence intervals is positive also a statistical analysis is p
275. iable r where fitting criteria and corresponding measurement values are set must be defined The order must be increasing and any decimal value within the integration interval is allowed The number and order of entries must coincide with the number and order of fitting functions of the 43 model function file and of course also with the number of measurement sets given Decimal numbers are rounded to the nearest integer that describes a line of the discretized system Method for Solving Discretized DAE The resulting system of differential algebraic equations is solved by an implicit Runge Kutta method of order 5 called RADAUS see Hairer and Wanner 199 The code is able to take algebraic equations into account and use dense output i e the integration is performed over the whole interval given by first and last time value and intermediate solution values are interpolated Gradients are obtained by external numerical differentiation Final Absolute Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the absolute global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible It is recommended to start with a relatively large accuracy e g 1 0E 6 together with a low number of iterations and to increase the accuracy when approaching a solution by restarts Final Relative Accuracy for
276. ial and algebraic equations must coincide with the number of model functions for the right hand side on the Fortran or PCOMP input file PDAE Model Parameters Si Number of Partial Differential Equations a Number of Partial Algebraic Equations 1 Flux in state equation no yes Number of Integration Areas 1 al Spatial Discretization Starting Value of Spatial Interval C 3 pt formula recursively for 2nd derivatives forward differences for first derivatives v 5 pt formula recursively for 2nd derivatives backward differences for first derivatives Order of spatial derivatives first second C 5 pt formula for 1st and 2nd derivatives C individual selection Ara PDAE Name Size Lines StatusL Status PB Discretization ODE Positions DAE Positions Fit Positions Di Accuracy absolute 1E 08 E Initial Stepsize 0 01 i Consistency Break Points Constraints Accuracy relative 1E 10 Jacobian bandwidth his Figure 7 21 PDAE Parameters Number of Integration Areas Partial differential equations may be defined in different areas Their total number is to be inserted here The number must coincide with the number of areas defined in the table that defines the individual area structure In each area a uniform discretization grid is applied Thus different areas with smooth transitions can be used to change the grid size Al Starting Value for Spatial Interval The spatia
277. ibed in Chapter 5 Model functions must be provided by the user either in form of the PCOMP language mentioned above or in form of Fortran code In the latter case the preparation of func tion and gradient values is described by initial comments of a code inserted by EASY FIT VodelDesion as a frame In case of PCOMP input the order in which variables and functions are to be inserted identifies their role in the mathematical model A full docu mentation of the model function input in this situation is presented in Chapter 6 The interactive system EAS Y FIT cPesis proceeds from a database for storing model information experimental data and results A complete context sensitive help option is included containing additional technical and organizational information about the input of data and optimization tolerances for example A brief outline of data organization and input is found in Chapter 7 The corresponding menu commands to define or alter data and functions to start an optimization run or to get reports on numerical results are described in Chapter 8 The numerical parameter estimation codes MODFIT and PDEFIT can be executed also outside of the interactive user interface A possible reason could be the solution of a large number of parameter estimation problems controlled by a separate command shell In this case a data input file is required that contains all information for starting the numerical algorithm The format of this fi
278. ight be desirable to add some randomly generated errors to available measurement values A typical situation arises when testing the identifiability of parame ters In this case one would generate some artificial measurements by a simulation run at predetermined parameter values A subsequent random perturbation and a data fitting test run from some other starting values indicates whether parameters can be identified or not There are two possibilities a Uniform distribution The user is asked to specify the percentage of a relative error that is to be added to the experimental value More precisely if an error of t is to be generated then each value say y is replaced by yi 1 t 2r 1 100 ri is a uniformly distributed random number between 0 and 1 b Normal distribution The user has to provide desired percentage of standard deviation variance t from which a normally distributed error is computed in the form where r is generated by the uniform normal distribution with mean 0 and variance 1 Since the original data are lost it is recommended to save the corresponding columns in the measurement table by the usual drag and drop action with respect to any other free column 18 EN Edit LINKER BAT jJ echo echo Copyright C 1995 2008 Klaus Schittkowski All rights reserved PRIN JJ rem LAHEY F77386 and F90 Linker rem rem BINXSBBLINIK EXE 951 1 LINK 2 AUX O
279. imental data in some others a large number of variables or system functions It is not reasonable to compile and link numerical codes with extraordinary large dimensioning parameters to serve all possible extreme situations Thus the dimensioning parameters can be adapted When activating the menu item an include file with all dimensioning parameters is opened for editing The meaning of the parameters is explained by initial comments also the default values can be retrieved from that file The user has the option compile and link FORTRAN codes directly from the corresponding form The include file can be saved under a separate name or on a separate directory 16 Edit COMPILE BAT rem LAHEY F77386 and F90 Compiler rem rem 953BINXF77L3 951 FOR NW rem pause KEM m rem SALFORD FTN 7 Compiler rem rem 3FTN77 1 SILENT AVINDOWS C OPTIMIZE rem 3FTN77 1 SILENT AVINDOWS BINARY DEBUG 1 0BJ rem pause KEM rem WATCOM F77 Compiler rem rem 3BINWWFC386p NOREF FPIG7 4 F O 1 0BJ 1 FOR rem Compaq Visual Fortran Compiler rem rem df optimize 5 fast traceback Awarn uninitialized c 1 for rem pause df fast check nobounds traceback c 961 for rem df Ars fast check bounds traceback Awarn uninitialized Awinapp debug pdbfile 1 c 1 for rem df traceback debug pdbfile 951 fc 951 for YEM Y rem AB
280. imes different from the one found in the reference We summarize a few characteristic data and the application background of the test problems that are available on the CD ROM from where further details can be retrieved Besides of problem name and some figures characterizing problem size we present also information how measurement data are obtained E Experimental data from literature or private communication S0 simulation without error S05 simulation with uniformly distributed error of 0 5 96 S1 simulation with uniformly distributed error of 1 S5 simulation with uniformly distributed error of 5 S10 simulation with uniformly distributed error of 10 S50 simulation with uniformly distributed error of 50 96 N1 simulation with normally distributed error c 1 N5 simulation with normally distributed error o 5 X comparison with exact solution ED experimental design none no experimental data set for example least squares test problem The difference between simulated and experimental data is that exact parameter values are 1 known in the first case Besides of a large collection of problems with practical experimental date there are also a few others where the data are constructed for instance in case of some optimal control problems or are determined more or less by hand In many other situations the exact solution of the differential equation is known and used to simulate experimental
281. in detail among the initial comments of the Fortran code of MODFIT Numerical results are stored on a file with extension RES and are read by EASY FIT Yo iDesi n as soon as numerical execution is terminated In case of a simulation run with a positive significance level statistical data are written to a file with extension STA in abbreviated form The code distributed together with EASY FIT Pesis allows the input of model functions on a user provided file with name lt MODEL gt FUN in a directory specified in the first line of MODFIT DAT In this case the input format is the PCOMP language must be used as outlined in the previous chapters A specific advantage is the automatic evaluation of derivatives There is no further compilation or link necessary and MODFIT can be started immediately as soon as both input files are created On the other hand a user has the option to implement all model functions in form of Fortran code and to link his own module to the object file of MODFIT Information about the remaining files to be linked in this case is found among the initial comments of the file MODFIT FOR that contains the main program The model data i e fitting criterion system equations bounds and initial values must be provided by a user defined subroutine called SYSFUN SUBROUTINE SYSFUN NP MAXP NO MAXO NF MAXF NR MAXR X Y T C YP Y0 FIT G DYP DY0 DFIT DG IFLAG 8 The meaning of the arguments is as follows NP
282. ind direction is determined internally The simple upwind scheme and the second and third order schemes can be applied only to scalar equations In case of the ENO scheme an additional explicit Runge Kutta methods is implemented to satisfy a so called CFL stability condition Area Data for Partial Differential Equations The structure the individual integration areas and therefore the positions of the transition equations are to be defined in form of a ta ble Note that only the status values are obligatory for each individual partial differential equation It is sufficient to define the corresponding information only within one line that identifies the area In this case the equation number must be 1 For each area and the differential equation identified by a serial number the following data must be set 39 Name The area may be characterized by an arbitrary string Size The length of the spatial interval of the corresponding area must be given Lines Within each area an equidistant grid with respect to the spatial variable is used to transform the PDE into a system of ordinary equations The number of grid points must be odd to get an even number of equidistant intervals for applying Simpson s rule in case of numerical integration with respect to spatial variable Note that the larger the number of grid points is the larger is the resulting ODE Status A status number identifies the type of the transition condition be tween a
283. ing Biochemistry Vol 93 9425 9430 Wang H Al Lawatia M Sharpley R C 1999 A characteristic domain decomposition and space time local refinement method for first order linear hyperbolic equations with interface Numerical Methods for Partial Differential Equations Vol 15 No 1 1 28 Wang Z Richards B E 1991 High resolution schemes for steady flow computation Journal of Computational Physics Vol 97 53 72 Wansbrough R W 1985 Modeling chemical reactors Chemical Engineering Vol 5 95 102 Watts D G 1981 An introduction to nonlinear least squares in Kinetic Data Analysis Design and Analysis of Enzyme and Pharmacokinetic Experiments L Endrenyi ed Plenum Press New York 1 24 Weinreb A Bryson A E Jr 1985 Optimal control of systems with hard control bounds IEEE Transactions on Automatic Control Vol AC 30 1135 1138 Weizhong D Nassar R 1999 A finite difference scheme for solving the heat transport equation at the microscale Numerical Methods for Partial Differential Equations Vol 15 No 6 697 708 Wen C S Yen T F 1977 Optimization of oil shale pyrolysis Chemical Engineering Sci ences Vol 32 346 349 36 547 548 549 590 551 552 553 554 555 556 557 558 559 560 561 Williams M L Landel R F Ferry J D 1955 The temperature dependence of relaxation mechanisms in amorphous polymers and other glass forming liquids Journal
284. ing formulations 2 32 2 33 and 2 34 respectively These restrictions define certain addi tional conditions to be satisfied for the parameters to be estimated Functions gi p 9m p must be continuously differentiable everywhere in the JR A typical situation is discussed in Example 2 1 where the exact satisfaction of the first and last fit are required The generation of more systematically introduced constraints will be discussed in subsequent sections Another frequent situation arises if for example the parameters to be estimated describe certain concentrations of fractions of a given amount of mass distribution so that the sum over all parameters must be one at least at an optimal solution 3 Example 2 11 POPUL Woe consider population dynamics of 10 species with a given maximum population rate each described by an ordinary differential equation Yi Gili Umaz Yi Biyi 2 51 fori 1 10 with Ymar 200 Initial values are yb piNo where p is an unknown fraction of the initial population of No 1 000 individuals for all species i 1 10 We are able to measure the total number of individual members of the population i e fitting criterion is h p y p t t gt p t There is no additional concentration parameter and only one measurement set Since each parameter p stands only for a fraction of a given constant value the single equality constraint is 10 g p X pi 1 0 i l W
285. inger 694 702 Molander M 1990 Computer aided modelling of distributed parameter process Technical Report No 193 School of Electrical and Computer Engineering Chalmers University of Technology G teborg Sweden Mor J J 1977 The Levenberg Marquardt algorithm implementation and theory in Nu merical Analysis G Watson ed Lecture Notes in Mathematics Vol 630 Springer Berlin Mor J J Garbow B S Hillstrom K E 1981 Testing unconstrained optimization software ACM Transactions on Mathematical Software Vol 7 No 1 17 41 Morrison R T Boyd R N 1983 Organic Chemistry Allyn and Bacon Morton K W Mayers D F 1994 Numerical Solution of Partial Differential Equations Cambridge University Press Munack A 1995 Simulation bioverfahrenstechnischer Prozesse in Prozessimulation H Schuler ed VCH Weinheim 409 455 Munack A Posten C 1989 Design of optimal dynamical experiments for parameter esti mation Proceedings of the Americal Control Conference Vol 4 2010 2016 M ller T G Noykova N Gyllenberg M Timmer J 2002 Parameter identification in dy namical models of anaerobic wastewater treatment Mathematical Biosciences Vol 177 178 147 160 Murray J D 1990 Mathematical Biology Springer New York 23 348 349 390 351 352 353 354 355 356 357 358 359 360 361 362 363 Naguma J Arimoto S Yoshizawa 1962 An
286. investigating stability or identifiability of a certain model before inserting experimental data it might be desirable to perform some preliminary tests with artificial measurement values The user has to provide the desired type of distribution the final time and the number of time values to be generated Note that the zero time is always generated and not counted If n is the number of time points until T equidistant time values are t T and exponential time value with shift s are given by the formula f g p9xP 57 1 exp s 1 for i Lim Generating of Measurements When investigating stability or identifiability of a certain model before inserting experimental data it might be desirable to perform some preliminary tests with artificial measurement values Proceeding from a previously performed simulation run it is possible to let the simulation results be inserted in the table containing the parameter estimation data Subsequently the given starting values for parameters to be estimated can be disturbed and a parameter estimation run can be started from this initial point Note that random errors should be added to the generated measurements to simulate experimental real life situations To generate simulated measurement data one has to define time values and if necessary also concentration values furthermore zero entries in the column that has to get the simulated measurement values and corresponding weights In many cases it m
287. ion when a variable switching point passes or approximates a measurement time value during an optimization run If both coincide and if there is a non continuous transition then the underlying model function is no longer differentiable with respect to the parameter to be optimized Possible reactions of the least squares algorithm are slow final convergence rates 62 or break downs because of internal numerical difficulties On the other hand variable switch ing points are very valuable when trying to model the input feed of chemical or biological processes given by a bang bang control function or any other one with variable break points Example 2 25 HEAT_B To show that our approach allows simultaneous fit of parame ters defining non continuous transitions with respect to time and space variables we extend Example 2 23 First we get the time dependent partial differential equation in two areas Ue 05 05 uS I 3 1 us Ju 2 112 2 _ 2 ul Du see also 2 104 with diffusion coefficients D and D gt and t gt 0 The initial heat distribution for t 0 is the same as before but a switching point T is introduced where we assume that a certain value a is to be added to the system i e ul p oz 0 asin zz 0 lt x lt 0 5 u p z 0 asin rx Unam SCH a 2 113 ul pz r wu p oz on o 0 lt zr lt 05 pozo wi pz n to 05 lt xr lt l see 2 105 where ul p x t and u p x t denote the solution in the
288. ional parameters with very severe internal dependencies 3 10 Yi Vkukioyi D y1 0 Di 0 1 D gt 3 11 Ya kyukigyi koi 2k22 y2 ya 0 0 The same statistical analysis as above leads to the significance intervals of Table 3 4 The correlation coefficients between ky and kis between ko and ko2 and between D and D are exactly 1 By successive elimination of parameters with highest coefficient of the eigenvector belonging to the lowest eigenvalue see Table 3 3 priority levels are computed as shown in 86 Linear Compartment Model 120 100 80 60 40 20 0 10 20 30 40 50 60 t Figure 3 2 Model Functions and Data Table 3 4 They exactly reflect the artificially generated dependencies The parameters ki k and D obtained the highest scores and are considered as the most significant ones We even observe that the influence of ko on the solution is greater than that of ko as can be expected from the different coefficients in 3 11 An important side effect is that the maximum correlation is reduced from 1 0 to 0 56 Besides of detecting dependencies among parameters the proposed analysis helps to find redundant ones as shown by a slight modification of 3 10 An additional redundant para meter r is added to the first differential equation leading to a very small perturbation of the solution by choosing e 107 Ah kiy er ml zs D Ui 1Y1
289. ions defined by flux functions the following formulae are available forward differences for first derivatives backward differences for first derivatives simple upwind formula second order scheme third order upwind biased scheme ENO scheme 34 PDE Model Parameters x Number of Partial Differential Equations 3 Spatial Discretization 5 A A Number of Integration Areas 1 C 3 pt formula recursively for 2nd derivatives Slang Value er Spell Inienval 9 C 5t formula recursively for 2nd derivatives C simple upwind formular scalar Ze 5 pt formula for 1st and 2nd derivatives 2nd order scheme scalar Order of spatial derivatives first second C forward differences for first derivatives C 3rd order scheme backward differences for first derivatives ENO scheme systems Flux in state equation no yes C individual selection Area PDE Name Size Lines StatusL Status H Discretization DDE Positions Fit Positions area implicit explicit Break Points Constraints Accuracy absolute 1E 07 al Initial Stepsize 0 01 p ps Accuracy relative 1E 07 8 Jacobian bandwidth pl D Integration Method Figure 7 18 PDE Parameters The forward and backward schemes can also be used to discretize individual scalar equations out of a system of multiple equations either only hyperbolic or mixed ones However the wind direction must be known a priori In all other case the upw
290. is of initial value problems with applications to shooting techniques DFG SPP Report No 403 Mathematisches Institut TU M nchen Bulirsch R 1971 Die Mehrzielmethode zur numerischen L sung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung Technical Report Carl Cranz Gesellschaft Oberpfaffenhofen Bulirsch R Kraft D 1994 Computational Optimal Control International Series of Nu merical Mathematics Vol 111 Birkh user Boston Basel Berlin Butcher J C 1963 Coefficients for the Study of Runge Kutta integration processes Journal of the Australian Mathematical Society Vol 3 185 201 Butcher J C 1964 Integration processes based on Radau quadrature formulas Mathematics of Computations Vol 18 233 244 Buwalda J G Ross G J S Stribley D B Tinker P B 1982 The development of endomy corrhizal root systems New Phytologist Vol 92 391 399 Buzzi Ferraris G Facchi G Forzetti P Troncani E 1984 Control optimization of tubular catalytic decay Industrial Engineering in Chemistry Vol 23 126 131 Buzzi Ferraris G Morbidelli M Forzetti P Carra S 1984 Deactivation of catalyst mathematical models for the control and optimization of reactors International Chemical Engineering Vol 24 441 451 Bykov V Yablonskii G Kim V 1978 On the simple model of kinetic self oscillations in catalitic reaction of CO oxidation Doklady AN USSR Vol 242 637 639 Byrne G D Hi
291. it direkter Suche Diploma Thesis Dept of Mathematics University of Bayreuth Germany Nishida N Ichikawa A Tazaki E 1972 Optimal design and control in a class of distributed parameter systems under uncertainty AIChE Journal Vol 18 561 568 Nocedal J Wright J 1999 Numerical Optimization Springer Series in Operational Re search Springer New York Nowak U 1995 A fully adaptive MOL treatment of parabolic 1D problems with extrapola tion techniques Preprint SC 95 25 ZIB Berlin Noykova N Miiller T G Gyllenberg M Timmer J 2001 Quantitative analysis of anaer obic wastewater treatment processes Identifiability and parameter estimation Biotechnology and Bioengineeriung Vol 78 89 103 Oberle H J 1987 Numerical Computation of Singular Control Functions for a Two Link Robot Arm Lecture Notes in Control and Information Sciences Vol 95 Springer Berlin 24 364 Odefey C Mayr L M Schmid F X 1995 Non prolyl cis trans peptide bond isomerization 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 as a rate determining step in protein unfolding and refolding Journal of Molecular Biology Vol 245 69 78 Ogden R W Saccomandi G Sgura I 2004 Fitting hyperelastic models to experimental data Computational Mechanics Vol 34 484 502 Oh S H Luus R 1975 Optimal feedback control of time delay systems AIChE Journal
292. ition conditions see 2 96 For the same reasons pointed out in Section 2 4 3 we suppose that n break or switching points with WELA STi et E 2 110 61 Linked Beams LESSEE SE 225 lt Z2z22ZZZZZZZ Po Figure 2 32 Third Order Transition Condition are given where T is the last experimental time value For the first integration interval the same initial boundary and transition values are given as before see 2 99 2 101 2 97 and 2 98 respectively But for all subsequent intervals the integration subject to the time variable is to be restarted at a switching point and new function values can be provided that may depend now also on the solution of the previous section Initial values at a switching point are evaluated from OM T Tk bi p ul p T Tk 1 v p Tk i x gt 2 111 v p Tk b p v p Tr 1 fori 1 ma and k 1 ny where ui p 7x1 and v p T 1 denote the solution of the coupled PDAE system in the previous time interval Since the right hand side of the partial differential equation 2 96 and also the corre sponding boundary and transition functions depend on the time variable they may change from one interval to the next Also non continuous transitions at switching points are al lowed It is possible that break points become variables to be adapted during the optimiza tion process as outlined in Example 2 8 However there exists a very dangerous situat
293. jonpuoo jeoq reourT uorjgpuoo TRUITT reourpuou AIST urop qo1d uorgjooApe reour uoryenbo uorjooAuoo uorsngtp reourT ondine uoryenbo ooe de T oqnj TEMI Y ur Aop 1eurure ApRoysuy uors1odsrp uorgooApe Aq uorjdaos orjourq uD e ur 9qOId e Surgeon SOURIQUIOUL 99 19 JIM sts zoudoluor Jo o13uoo Ted WOISIOA qorpun 214 oda oAIs1odstp oArjooApe 41odsuerj asnos oArjowoj ode DAISIOASIP 9A1J909APe 410dsuer o3n os DALJOCIH uorjd ros 10 urloq3ost IMUISURT YIM oueiquiour uSnorg j1odsueij uo uorsnpgrp Aq eue1quiour ysnoiyy 110dsuer uo uorjnpuoo jeoq ur uro qo id os1oAug ooejIo UI IA urojs4g popopour you ooejproqur YIM urojsKg uoryenbo erquod9grp o1Soqut IL sorureu Ap uorje idoq uondaos p uoryd rosqe uorsngrp s1oXe your YIM 3sKqe3e UIo38s S 3s93 ot oqed A punosbyo0q CO CC Ch Ch CC CH D co QN CH CH CC On Ch CH Ch Ch oooo CH es a Sco 6 So oO oo OUO Oc OC O t CO CH O G O Ch OC Ch CH un o E Cl On r Cl OO N N e NN c c on n ri amp VII 0c Asra GIG 04 GC OFT 06 001 0c OT TOT GG LT 0c ve 96 TOT 0 8I OT GS 42 06 ut SONA c ri o CQ CN CO CN C CN NOD o wt 2 o MAA N O N co OTXULTIVIN ONT SSW ISSO I CINOHHON I TNOYHON T dAH NIT Gel AH NIT IdAH NIT OH NTI ACV NIT dOG I dAOV IdV I MOTA NYT duos NIM NIM O LNOf CINSH LOSI DNH LOS ENVUL
294. l interval for the one dimensional space coor dinate is defined by the initial value to be defined here and the individual lengths of the areas defined in the corresponding input table Order of Partial Differential Equation If the right hand side of the partial differential equa tion depends only on first spatial derivatives it is not necessary to evaluate second order approximations for Uy Flux in state equation Flux functions facilitate the input of more complex equations If set to yes there must be variables identifying flux functions and their spatial derivatives in the PCOMP input file moreover a defining equation of the flux for each state variable Spatial Discretization Partial differential equations are discretized with respect to the spatial variable by the following difference formulae 3 point difference formula recursively for second derivatives 5 point difference formula recursively for second derivatives 5 point difference formula for first and second derivatives forward differences for first derivatives backward differences for first derivatives All of these difference formulae can individually be applied to the spatial derivatives of each state variable The forward and backward schemes can also be used to discretize individual scalar equations out of a system of multiple equations either only hyperbolic or mixed ones However the wind direction must be known a priori Area Data for Partial Differential Equati
295. lace Transformations For the execution of the numerical analysis program MODFIT for models in the Laplace space we need some further data that cannot be retrieved from the model function file or other information Number of Measurement Sets The number of measurement sets must coincide with the num ber of data sets as given in the input table for experimental data Number of Concentration Values The number of concentrations must coincide with the number of concentrations as given in the input table for experimental data If the value inserted is positive and the PCOMP input language is used then a concentration variable must be declared in the model function file If 1 is inserted it is supposed that the fitting criteria depend on an additional concentration variable and that one concentration value is assigned to each time value Number of Iterations for Back Transformations MODFIT is capable solve parameter esti mation problems where the model functions are defined in the Laplace space The quadra ture formula of Stehfest 490 is implemented to transform the function values from the Laplace space to the original space in the time variable A value of 5 or 6 is recommended to maximize accuracy of the approximation and to avoid numerical instabilities Laplace Model Parameters x A Number of Measurement Sets H L a Number of Concentrations 7 x Figure 7 10 Parameters for Laplace Equations 20 7 2 4 Model Dat
296. le A Wang T Melvin W W 1987 Optimal abort landing trajectories in the presence of windshear Journal of Optimization Theory and Applications Vol 12 815 821 22 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 Mishkin M A Saguy I Karel M 1982 Applications of optimisation in food dehydration Food Technology Vol 36 101 109 Mishkin M A 1983 Dynamic modeling simulation and optimization of quality changes in air drying of foodstuffs Ph D Thesis Massachusetts Institute of Technology Cambrigde MA USA Mishkin M A 1983 Dynamic optimization of dehydration processes Minimizing browning in dehydration of potatoes Journal of Food Science Vol 48 1617 1621 Missel P J 2000 Finite element modeling of diffusion and partioning in biological systems Report Drug Delivery Alcon Research Ltd Fort Worth USA Mittelmann H D 2001 Sufficient optimality for discretized parabolic and elliptic control problems in Fast Solution of Discretized Optimization Problems K H Hoffmann R H W Hoppe and V Schulz eds ISNM 138 Birkhauser Basel Mittra R 1973 Computer Techniques for Electromagnetics Pergamon Press Oxford Mohler R R Farooci Z Heilig T 1984 An immune lymphocyte circulation system in System Modelling and Optimization P Thoft Christensen ed Lecture Notes in Control and Information Sciences Vol 59 Spr
297. le for x and that we got positive values at t 0 In other words each of the five separate processes starts at an unknown initial time T lt 0 j 1 5 The first attempt could be to define additional optimization parameters for all initial values all together 15 additional optimization parameters The number of iterations ni the final residual value and the optimal parameter set are listed in Table 2 9 case 1 However the results are incorrect since we do not take into account that initial times for the three equations 2 68 must coincide for each concentration value Thus we suppose that there is one fixed initial time ty 0 1 The results are also shown in Table 2 9 case 2 Finally we introduce one variable initial time for each of the 5 con centrations and start the data fitting run at tT 0 1 j 1 5 The integration is initialized at t 0 2 Initial values at 0 2 and right hand side of corresponding equations are set to zero see also Figures 2 21 to 2 23 for plots of the state variables over time and concentration We know that initial values of the process under consideration are zero 39 ti Ti Ti2 Lig Lia Lis 0 0 014 0 016 0 015 0 009 0 007 0 1666 0 065 0 059 0 051 0 038 0 031 0 3333 0 117 0 100 0 088 0 069 0 055 0 5 0 167 0 142 0 123 0 099 0 080 0 6666 0 214 0 181 0 157 0 129 0 112 0 8333 0 264 0 220 0 193 0 159 0 137 1 0 311 0 261 0 228 0 193 0 166 Table 2 8 Experimental Data case na resi
298. le is documented in Chapter 9 in detail Usage of the codes is illustrated by a few test examples The database of the delivered EAS Y FIT s9 version contains 1 300 academic and real life examples i e test problems with some realistic practical background Application areas are pharmacy biochemistry chemical engineering and mechanical engineering The purpose for attaching a comprehensive collection of test problems in Chapter 10 is to become familiar with the PCOMP language and the implementation of a new model since a large variety of different model structures is offered The problems can be used for selecting a reference example when trying to install own dynamical models or to test the accuracy or efficiency of the algorithms available within EASY FITVodelDesign Chapter 2 Data Fitting Models Our goal is to estimate parameters in explicit model functions Laplace transforms steady state systems systems of ordinary differential equations systems of differential algebraic equations systems of one dimensional time dependent partial differential equations systems of one dimensional partial differential algebraic equations Proceeding from given experimental data i e observation times and measurements the minimum least squares distance of measured data from a fitting criterion is to be computed that depends on the solution of the dynamical system In this chapter we summarize in detail how the model functi
299. least squares problems in differential algebraic equations Proceedings of the Fourth Seminar NUMDIFF 4 Halle Numerical Treatment of Differential Equations Teubner Texte zur Mathematik Vol 104 Teubner Stuttgart Boderke P Schittkowski K Wolf M Merkle H P 2000 A mathematical model for diffusion and concurrent metabolism in metabolically active tissue Journal on Theoretical Biology Vol 204 No 3 393 407 Bojkov B Hansel R Luus R 1993 Application of direct search optimization to optimal control problems Hungarian Journal of Industrial Chemistry Vol 21 177 185 Borggaard J Burns J 1997 A PDE sensitivity method for optimal aerodynamic design Journal of Computational Physics Vol 136 No 2 366 384 4 57 58 59 60 61 62 Ee 64 65 66 67 68 69 70 71 72 Bossel H 1992 Modellbildung und Simulation Vieweg Braunschweig Box G P Hunter W G MacGregor J F Erjavec J 1973 Some problems associated with the analysis of multiresponse data Technometrics Vol 15 33 51 Box G P Hunter W G Hunter J S 1978 Statistics for Experimenters John Wiley New York Bryson A E Denham W F Dreyfus S E 1963 Optimal programming problems with in equality constraints AIAA Journal Vol 1 No 11 2544 2550 Bryson A E Ho Y C 1975 Applied Optimal Control Hemisphere New York Buchauer O Hiltmann P Kiehl M 1992 Sensitivity analys
300. liar with the underlying mathematical theory to understand the data in detail For the meaning of the parameters displayed it is necessary to read the corresponding user guides If the internal editor is used and if MODFIT or PDEFIT generate too much output it is possible that the size of the text size extends 32 K To avoid an internal error situation please switch to another editor in this case for example to EDITOR EXE 8 5 Data Command The menu command offers a few flexible opportunities for im and export of measurement data l File Edit Start Report Delete Make Utilities Read measurement data Export to EXCEL Import from EXCEL Model o Information Diffusion of water through soil convection and Project Number Demo Unit for 4 User Name Schitkowski Unit for AM Figure 8 7 Import of Experimental Data Experimental data can be read from any text file in standard format where the corre sponding time concentration measurement and weight data must be organized in exactly the same order in which they are used in the input table of EASY FITYodelDesign There is no special format required for real numbers in the input file but decimal or exponential numbers are not to be separated by anything else than a blank or new line Alternatively data may be organized in rows possessing identical structure where the initial column position and the length of an item to be read must be known
301. licit simultaneous sensitivity analysis ACM Transactions on Mathematical Software Vol 14 No 2 61 67 Levenberg K 1944 A method for the solution of certain problems in least squares Quarterly Applied Mathematics Vol 2 164 168 Lewis R M Patera A T Peraire J 2000 A posteriori finite element bounds for sensitivity derivatives of partial differential equation outputs Finite Elements in Design Vol 34 271 290 Lewis J W 1994 Modeling Engineeing Systems LLH Technology Publishing 1994 Liepelt M Schittkowski K 2000 Algorithm 746 New features of PCOMP a FORTRAN code for automatic differentiation ACM Transactions on Mathematical Software Vol 26 No 3 352 362 Liepelt M Schittkowski K 2000 Optimal Control of Distributed Systems with Break Points in Online Optimization of Large Scale Systems M Gr tschel S O Krumke J Rambau eds Springer Berlin 271 294 Lindberg P O Wolf A 1998 Optimization of the short term operation of a cascade of hydro power stations in Optimal Contol Theory Algorithms and Applications W W Hager P M Padalos eds Kluwer Academic Publishers Dordrecht Boston London 326 345 Lindstr m P 1982 A stabilized Gaup Newton algorithm for unconstrained least squares problems Report UMINF 102 82 Institute of Information Processing University of Umea Umea Sweden 19 288 289 290 291 292 293 294 295 296 297 298 299
302. lots are to be generated for state variables for which E data do not exist The subsequent table contains the actual number l lt l of terms taken into account in the final least squares formulation The data fitting function bin y p t c z p t c t c depends on a concentration parame ter c and in addition on the solution y p t c and z p t c of a system of ma differential and ma algebraic equations Yi Fi p Y Ztc yi 0 zx y p c D Uma Fmalp Y z t c Um L0 yh D C 0 Gilo y 2 t c z1 0 zt c H 0 Ga DU Z t C H E Without loss of generality we assume that the initial time is zero Now y x t c and z z t c are solution vectors of a joint system of ma m differential and algebraic equations DAE The initial values of the differential equation system y p c ur p c and 21 p c zb p c may depend on one or more of the system parameters to be estimated and on the concentration parameter c The system of differential equations is called an index 1 problem or an index 1 DAE if the algebraic equations can be solved with respect to z i e if the matrix V G p u z 0 6 possesses full rank In this case consistent initial values are can be computed internally The resulting parameter estimation problems is min 2 3 xen xu wh hip y p ti Ge z p ti eo ti cj ys g p 0 E 1 2 Me gt ne Hd f gp 0 j m 1 m pi lt p lt p We as
303. low of parser syntax error available memory exceeded index or index set not allowed error during dynamic storage allocation wrong number of indices wrong number of arguments number of variables different from declaration number of functions different from declaration END sign not allowed Fortran code exceeds line domain error bad input format length of working array IWA too small length of working array WA too small ATANH domain error LOG domain error SQRT domain error ASIN domain error ACOS domain error ACOSH domain error LABEL defined more than once LABEL not found wrong index expression wrong call of the subroutine SYMINP wrong call of the subroutine SYMPRP compilation of the source file in GRAD mode interpolation values not in right order not enough space for interpolation functions in subroutine REVCDE length of working array IWA in subroutine SYMFOR too small not enough interpolation values 16 Chapter 6 Model Functions and Equations Model functions of the test examples are defined in the PCOMP language The meaning of variables and functions is fixed by their serial order Identifiers can be chosen arbitrarily 6 1 Explicit Model Functions To define model variables and explicit fitting functions in the PCOMP syntax one has to follow certain guidelines for the declaration of parameters and functions since the order in which these items are defined is essential for the interface be
304. lso as our fitting criterion Parameters to be estimated are L and k The corresponding PCOMP input file is Q l C C Problem HEAT C e l 1 C REAL CONSTANT pi 3 1415926535 C VARIABLE L k Uy Wek Xx vj x t C FUNCTION u t ut uxx C FUNCTION uO u0 DSIN pi x L C FUNCTION v_t v_t k pi L DEXP pi L 2 t C FUNCTION vO vO k L pi C FUNCTION u_left u_left 0 C FUNCTION u_right u_right 0 C FUNCTION h h v C END C 12 6 7 Partial Differential Algebraic Equations Very similar to the definition of data fitting problems based on partial differential equations outlined in the previous section we have to define fitting criteria differential equations initial and boundary conditions coupling and transition equations and constraints in a suitable format For defining variables we need the following rules 1 8 The first names are the identifiers for n independent parameters to be estimated pi iau bes The subsequent names identify the n state variables of the system u1 Un Where first the differential then the algebraic variables must be listed In a similar way the corresponding variables denoting the first and second spatial derivatives of differential and algebraic variables are to be declared in this order uiz Vp nas ANd Wipes oiy ies
305. lt lt CH GO Cl Na oO cO CO CH No c8 8y TOT VI OV 0G 001 OT GI v9 OV Foi 8 0 0 9 c9 08 TOT 99 IG 09 o 1 ON HN aO 10 rd C2 1 O MAN C aO me OT ey LXHUY dNALATY Vuldsadu ANIL SHY TAOTdHYA Odd AV IWH SAY DAY DOIILTTH NOLLOVWH HOANOVAY UL OVA OVA AULNG SH ULSO qu ILVO Od GIOSLVU TTOS LVU COW ALVY dNVa Liddvu SATOUAd NICIYAd VOZOLOUd NIALOUd IWDU 36 p nurmuoo SS SS SS SS IS gS SS SS gS gS gS uou 9uou IS 9uou SS SS uou SS SS uou 020p sog 80 86 LVG vez vez vez TES VEG GOS vez GEG OLE SOP FG EHS Kn oUILARISOULIOY Aq oSpn s 1oyVMoyseAM JO uorjrsodurooop euro T ds pue soouongur Ju trsu yya uro3s4s Surje s Add P9ZIJ99SIP uorsngrp peuriopsue Jopour uorge ndod o duitg urojs s eorSo ooo o duitg Sojeulro1e JO UOI32 91 PELNU O doous ur umrisoeoorper Jo JIOASULAT SOU dro suo dreys UOISSTUISUB 1 PUL IOJZIOUL OC Aq UAP qU 1oxeug SISATeUR JIATyISUOS BUTJSO 10g uro qo id 3s93 oruropeoe HIIS 1039891 SNONUTJUOD IWas ur suorgoeor erguonbog 1032 91 23eq ruros ur uoryego1sos YPM uorjoeor o duutg 1039891 SnOnUIjUOO IUIOS Y UI SUOI32 91 9 Te q SUVS oi podde pout opuopodo YTAS Uorjoeo1 eoruroqo Ip SUIS uio qoud o1juoo repnsutg SUT OOD YPM 1039891 qojeq ruies Jo o1juoo peurrd sorno
306. lternatively the user can choose between the following two high resolution schemes APFLUX 2 0 first order upwind scheme APFLUX 3 0 second order central differences If 11 0 lt 99 1 the system of PDE s is supposed to consist of advection equations of the form Ou Ot Of p u Ox plus inhomogeneous part to be discretized by the ENO method with APFLUX IJ I Approximating polynomial order for state variable u at cell wall by Marquina s rule J Approximating polynomial order for numerical flux by ENO Roe rule METHOD Choice of analysis or optimization algorithm NORM METHOD 0 Simulation NUMGRA METHOD 1 Call of DENLP Schittkowski 429 METHOD 2 dummy NORM determines the data fitting norm NORM 1 Li norm sum of absolute residuals NORM 2 L norm sum of squared residuals 32 NORM 3 Lo norm maximum of absolute residuals L1 and maximum norm are applicable only for a sim ulation run or a data fitting run with DFNLP NUMGRA must be set for gradient evaluation NUMGRA 1 analytical derivatives available NUMGRA 0 1 forward differences NUMGRA 2 two sided differences NUMGRA 3 5 point difference formula a6 4x 15 OPTP1 Maximum number of iterations for the chosen optimiza tion algorithm In case of a simulation run the desired significance level 1 5 or 10 is to be inserted H H a6 4x 15 OPTP2 Additional parameter for chosen optimization algo rithm METHOD 1 maxi
307. ltibody system Journal of Computational and Applied Mathematics Vol 50 523 532 Sincovec R F Madsen N K 1975 Software for nonlinear partial differential equations ACM Transactions on Mathematical Software Vol 1 No 3 232 260 Slider H C 1976 Practical Petroleum Reservoir Engineering Methods The Petroleum Pub lishing Company Tulsa Oklahoma Smith G D 1985 Numerical Solution of Partial Differential Equations Finite Difference Methods Clarendon Press Oxford Applied Mathematics and Computing Science Series Smith M G 1966 Laplace Transform Theory Van Nostrand Smoller J 1994 Shock Waves and Reaction Diffusion Equations Grundlehren der mathe matischen Wissenschaften Vol 258 Springer Berlin Spellucci P 1993 Numerische Verfahren der nichtlinearen Optimierung Birkhauser Boston Basel Berlin Spellucci P 1998 A SQP method for general nonlinear programs using only equality con strained subproblems Mathematical Programming Vol 82 413 448 32 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 Spicer P T Pratsinis S E 1996 Coagulation and fragmentation universal steady state particle size distribution AICHE Journal Vol 42 No 6 1612 1620 Spiegel M R 1965 Laplace Transforms Schaum s Outline Series McGraw Hill New York Spoelstra J van Wyk D J 1987 A method of solution for a non linear diffusi
308. lysis kinetics Histochemistry Vol 60 53 60 Pohjanpalo H 1978 System identifiability based on power series expansion of the solution Mathematical Bioscience Vol 41 21 33 Posten C Munack A 1989 On line application of parameter estimation accuracy to biotechnical processes Proceedings of the Americal Control Conference Vol 3 2181 2186 Powell M J D 1978 A fast algorithm for nonlinearly constraint optimization calculations in Numerical Analysis G A Watson ed Lecture Notes in Mathematics Vol 630 Springer Berlin Powell M J D 1978 The convergence of variable metric methods for nonlinearly constrained optimization calculations in Nonlinear Programming 3 O L Mangasarian R R Meyer S M Robinson eds Academic Press New York London Pratt W B Taylor P 1990 Principles of Drug Action Churchill Livingstone New York Preston A J Berzins M 1991 Algorithms for the location of discintinuities in dynamic simulation problems Computers and Chemical Engineering Vol 15 701 713 Price H Varga R Warren J 1966 Application of oscillation matrices to diffusion convection equations Journal of Methematical Physics 301 311 R umsch ssel S 1998 Rechnerunterst tzte Vorverarbeitung und Codierung verfahrenstech nischer Modelle f r die Simulationsumgebung DIVA Fortschrittberichte VDI Reihe 20 Nr 270 VDI D sseldorf Radu F A Bause M Knabner P Lee G W Friess W C 2001 Drug relea
309. m he dra 89 3 4 Experimental Design with Weights 101 Numerical Algorithms 1 41 Data Fitting Algorithms 4 ek EE AN ss sq o nE ee ee g k E 1 42 Steady State Systems o se cs ek s ws e w h w W e a e RR E e w R w W W 2 4 3 Laplace Back Transformation 3 44 Ordinary Diferential Equations o se lt s e EIN sas ee soos Oe w h w w N 4 4 5 Differential Algebraic Equations 5 46 Partial Differential Equations lt 4 464346 844 ow eb w w Q w be w NW 6 4 7 Partial Differential Algebraic Equations aaa len 9 0 AS AE AUS ia No S sus e d ue a Ec e DELE E d 1 The Modeling Language PCOMP 1 5 1 Automatic Differentiation 2252s sk o RR RE RC 1 52 Input Formal forPCOMP 2 o 26424 64 824 oe ee k s XR ee s a 4 5 3 Error Messages of PCOMP sn cx LS Model Functions and Equations 1 bl Esplicit Model Functions gt lt s soe rer ea w Sow sut dear ES RE S 1 02 Laplace Transiormstione s scos s s hirs eg hira X3 9E sS er S 3 6 3 Systems of Steady State Equations e eee 4 64 Ordinary Diferential Equations oe s s soss s sa 84 244 ee Gorda 6 6 5 Differential Algebraic Equations 8 6 6 Time Dependent Partial Differential Equations 0 6 7 Partial Differential Algebraic Equations lll n 3 Re a Problem Data and Solution Tolerances 1 7 1 Model Independent Information d 7 1 1 Problem Name 22122 EE a ok m s a e AN x 1 7 1 2 Documentation Text
310. matrix Os t V s p z t c AS 2j 2 23 i 1 m j 1 m has full rank for all p with p lt p lt p and for all z for which a solution z p t c exists Consequently the function z p t c is differentiable with respect to all p in the feasible domain Now let t be fixed and let z p t c a solution of the system of equations If we de note s p z t c s1 p z t c Sm p 2 t c for all x and z we get from the identity s p z p t c t c 0 which is to be satisfied for all p the derivative Vos p z p t e t c E Vz p t c Vzs p z p t t c 0 2 24 Here V s p z t c and V s p z t c denote the Jacobian matrices of the vector valued function s p z t c with respect to the parameters p and z respectively In other words the desired Jacobian Vz p t c is obtained by solving the linear system Vps p z p t c t c VV s p z p t c t c 0 2 25 11 where V is am x n matrix Note that we describe here the implicit function theorem Since V 5 p z p t c t c is nonsingular the above system is uniquely solvable Finally we obtain the gradients of the fitting criterion from V fU E 205 V h p z p 156 tas kn V gp ti c V hip z p tc tae 2 26 fori 1 4 j7 1 l and k 1 r where z p t c is the solution of the system of nonlinear equations 2 22 and Vz p t c V computed from 2 25 In addition we allow any nonlinear restrictions on the parameters to be estimate
311. merical approximation of gradients the differential equation must be solved as accurately as possible It is recommended to start with a relatively large accuracy e g 1 0E 6 together with a low number of iterations and to increase the accuracy when approaching a solution by restarts The parameter is not needed for the explicit solver in case of an ENO scheme In case of internal selection of a perturbation tolerance for computing numerical derivatives however the parameter serves as a guess for the accuracy by which function values are provided Final Relative Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the relative global error is to be inserted In case of numerical approximation of gradients the differential equation must be solved as accurately as possible Again it is recommended to start with a relatively large accuracy say 1 0E 6 and a low number of iterations and to increase the accuracy when approaching a solution by restarts For the explicit solver with fixed steplength needed for ENO schemes a reduction factor is to be specified by which the given stepsize is scaled if the CFL condition is not satisfied Initial Stepsize for Solving Differential Equations Define initial stepsize for differential equa tion method used The parameter is adapted rapidly by internal steplength calculation Bandwidth of Jacobian of Right Hand Side Implicit methods requir
312. meters requiring a large number of steps e badly scaled model functions and in particular measurement values e non differentiable model functions We have to know that all efficient optimization algorithms developed for the problem class we are considering require differentiable fitting criteria and the availability of a starting point from which the iteration cycle is initiated Additional difficulties arise in the presence of nonlinear constraints in particular if they are badly stated ill conditioned badly scaled linearly dependent or worst of all contradictory Thus users of parameter estimation software are often faced with the situation that the algorithm is unable to get a satisfactory return subject to the given solution tolerances and that one has to restart the solution cycle by changing tolerances internal algorithmic decisions or at least the starting values to get a better result To sum up a black box approach to solve a parameter estimation problem does not exist and a typical life cycle of a solution process consists of stepwise redesign of solution data The parameter estimation problem alternative phrases are data fitting or system identi fication is outlined in Chapter 2 Is is shown how the dynamical systems have to be adapted to fit into the least squares formulation required for starting an optimization algorithm A first question is always how to get suitable confidence intervals for the estimated paramete
313. mming algorithm with distributed and non monotone line search user s guide version 2 2 Report Department of Computer Science University of Bayreuth Schittkowski K 2007 Experimental design tools for ordinary and algebraic differential equations Industrial and Engineering Chemistry Research Vol 46 9137 9147 Schittkowski K 2007 NLPQLB A Fortran implementation of an SQP algorithm with with active set strategy for solving optimization problems with a very large number of constraints user s guide version 2 0 Report Department of Computer Science University of Bayreuth 29 443 444 445 446 447 448 449 450 451 452 453 454 455 456 Schittkowski K 2007 An active set strategy for solving optimization problems with up to 60 000 000 nonlinear constraints submitted for publication Schittkowski K 2007 Parameter identification in one dimensional partial differential alge braic equations GAMM Mitteilungen Vol 30 No 2 352 375 Schittkowski K 2007 NLPMMX A Fortran implementation of an SQP algorithm for min max optimization user s guide version 1 0 Report Department of Computer Science University of Bayreuth Schittkowski K 2007 NLPLSQ A Fortran implementation of an SQP Gauss Newton al gorithm for least squares optimization user s guide version 1 0 Report Department of Computer Science University of Bayreuth Schittkowski K 2008 Pa
314. more than 500 experimental data i e more than 1 000 nonlinear constraints The modified SQP code is called NLPQLB see Schittkowski 442 Experimental Design Experimental design helps to find suitable values for additional so called design parameters for example initial concentrations or input feeds which have to be set before conducting experiments Various experimental design methods have been dis cussed in the literature before see e g Winer Brown and Michels 551 Ryan 414 Rudolph and Herrend rfer 412 Baltes et al 19 or Lohmann et al 296 Since the confidence in Data Fitting Tolerances X A Computation Starting at Number of Iterations 100 3 Output Flag one line per iteration Last Computed Parameter Set Gradient Evaluation forward differences _ gt A Order of Numerical Differentiation Tolerance 0 4 A Termination Tolerance 1E 07 E DN a A Final Residual Estimate 0 01 A Confidence Level 5 4 Figure 7 3 Data Fitting Tolerances tervals mentioned above are mainly determined by the diagonal elements of the covariance matrix a possible objective function is the trace of this matrix see Schittkowski 441 for more details and a couple of case studies A further option is to locate experimental time values Especially in case of time expensive experiments it is highly desirable to minimize their number and to conduct experiments only within relevant time intervals Thus we
315. mulated as an additional nonlinear equality constraint is to be satisfied at the optimal solution In the first two cases the dynamical system can be expected to have a unique solution so that both boundary conditions are satisfied when evaluating the fitting criterion whereas in the second case fulfillment of the right boundary condition is not guaranteed and can be expected at most at an optimal solution For more details about boundary value problems BVP and their numerical solution see Ascher Mattheij and Russel 10 Mattheij and Molnaar 324 or Ascher and Petzold 11 Let a system of second order ordinary differential equations with boundary values be given in explicit form ji Fi y Y t c gt y1 0 T in c yi T E yt p c p 2 64 s F p y Ue Ym 0 a Oe Ym T Ha 6 3 where we omit algebraic equations for simplicity There is no need to develop a special integration algorithm in case of a data fitting problem since the boundary value problem is 35 easily transformed into a first order initial value problem with additional free parameters to be optimized and additional state variables From 2 64 we get the equivalent system Yi U1 v1 0 ve Ym Um gt Um 0 Un gt 0 S 2 65 v PA y 0 6 y1 0 Yi p c gt Us F p y v t c gt Ym 0 Yin p c gt where vi t Um t are the additional state variables to eliminate second derivatives and v9 v9 additi
316. mum number of line search steps a6 4x 15 OPTP3 Output level of optimization algorithm chosen The output is directed to a file with extension HIS Only the residuals are displayed on screen a6 4x g10 4 OPTE1 Tolerance for chosen optimization algorithm METHOD 0 tolerance for rank determination METHOD _1 final termination tolerance a6 4x g10 4 OPTE2 Tolerance for chosen optimization algorithm METHOD 1 expected size of residual a6 4x g10 4 OPTE3 Tolerance for chosen optimization algorithm dummy ODEPL1 Parameter for selection of differential equation solver 33 ODEP1 1 dummy ODEP1 2 dummy ODEP1 3 dummy ODEP1 4 RADAUS implicit Runge Kutta order 5 ODEP1 5 dummy ODEP1 6 dummy ODEP1 7 TVDRK explicit Runge Kutta order 5 for systems of advection equations without inhomogeneous term 37 a6 4x i5 ODEP2 Order of used method only for implicit methods ODEP2 0 no derivatives ODEP2 1 derivatives of right hand side supplied 38 a6 4x 15 ODEP3 Approximate number of correct digits when gradients are evaluated numerically by forward differences If set to 0 the tolerance in internally computed 39 a6 4x 515 ODEP4 Bandwidth of Jacobian of right hand side only for im plicit solver must be smaller than number of ODE s of discretized system In case of ODEP4 0 usage of full matrix is assumed 40 a6 4x 10 4 ODEEI Final termination accuracy for ODE solver with res
317. n be derived also for nonlinear ones Under additional regularity assumptions see Seber and Wild 458 D and s s p l n are consistent estimates of p and o that means they converge with probability 1 to the true values and are asymptotically normally distributed as l goes to infinity Moreover we know that due to the normal distribution of the errors p is also a maximum likelihood estimator The error in parameters p p is approximately normally distributed with mean value 0 and covariance matrix o I l where I is defined by I F F In addition the expression 1 ns follows the F distribution with n l n degrees of freedom within the linearization error Thus an approximate 100 1 a confidence region for p is given by the set lp mo Ip znsF2 3 3 where Vf p Vf p estimates I This result is very similar to the corresponding confidence region for linear models For a numerical implementation however 3 3 is inconvenient To get individual confi dence intervals for the coefficients of p we consider an arbitrary linear combination a p It is possible to show that approximately 8 p T P p folium 3 4 where t n is the t distribution with n degrees of freedom A 100 1 a confidence interval is then given by ap tel sy aT I la a pt t2 sy aT fo 3 5 82 When setting a e for i 1 n successively where e is the i th unit vector
318. n ecological system used as a standard parameter estimation test problem in the literature cf Clark 93 Varah 522 or Edsberg and Wedin 130 The so called Lotka Volterra system consists of two equations Yi kiyi koyiya 2 59 Ya kaya kayiyo with initial values y1 0 0 4 and y2 0 1 Parameters to be estimated are ki ko ka and k4 If we insert the values ky ka ks 0 5 and ky 0 2 and try to integrate the system from t 0 to t 10 then every ODE solver must break down because of a singularity near jc Thus we have to apply the shooting technique for being able to avoid singularities If we assume that measurements for yj and ya are available in our case obtained by sim ulation subject to 10 time values t 0 1 t2 0 2 tio 1 the parameter values p ky ka kz ka 1 1 1 1 and a subsequent perturbation of 5 we get the least squares problem min Dir ya p ts 9D X alo ti 2 yi p ti sl 0 t ba 1 p R s1 52 IR 9 t 2 60 yo p ti s 0 l 32 Predator Prey Model Figure 2 17 Initial Multiple Shooting Trajectories Here we have s sl si and sg s2 S ll wi p t and yo p ti are the numerical solution of the Lotka Volterra equation of the proceeding interval from t 4 to ti In this case
319. n is always beginning at 0 Alternatively a fitting criterion might depend on time and concentration variables where each time value corresponds to one concentration value which may all be different This option is not available for models based on partial differential equations Column headed by conc entration Input of concentration values c There must be one and the same concentration value for one series of time values If more than one concentration value is available we assume that all sets proceed from the same series of time values and the same concentration value Otherwise experiments must be deleted by zero weights If a fitting criterion depends on time and concentration variables where each time value corresponds to one concentration value then a suitable concentration value must be assigned to each time value This option is not available for models based on partial differential equations Column headed by value Input of measurement value yr for the actual measurement set Any legal format for real numbers is allowed k Column headed by weight Input of nonnegative weight factor w for actual measurement set If the weight is set to zero the corresponding measurement value is not taken into account 7 1 11 Data Fitting Norm Experimental data can be fitted in three different norms L5 norm The classical least squares norm minimizes the sum of all squares of differences between model function and measurement val
320. n one experimental data set can be fitted within a model formulation 2 The fitting criteria are arbitrary functions depending on the parameters to be esti mated the solution of the underlying dynamical system and the time variable 3 The model may possess arbitrary equality or inequality constraints with respect to the parameters to be estimated and upper and lower bounds for the parameters 2 10 11 12 13 14 15 16 Model equations may contain an additional independent parameter for example ex perimental concentration or temperature values Differential algebraic equations can be solved up to index 3 Consistent initial values for index 1 formulations are computed internally In case of partial differential equations also coupled ordinary differential equations and non continuous transitions for state variable and flux between different areas can be taken into account Differential equation models may possess additional break or switching points where the model dynamics is changed and where integration is restarted for example if a new dose is applied in case of a pharmacokinetic model The switching points mentioned before may become optimization variables to allow the modeling of dynamical input for instance to compute optimal bang bang feed controls of a chemical reactor The model functions may be defined by their Laplace transforms where the back transformation is perf
321. na 8V Id LVIA 9V Id Sv Id VV IG IWDU 26 p nuruoo IS SS SS SS a SS gS gS SS SS gS SS SS X IN SS a 0S a a a GS SS a GS GS gS 030p Log 91 9cT 946 vVE WE LGE 9cT Du Du 897 T9G EZS 9cT gel VEG VEG VEG fas uorjejuourl9J Jo o13uoo o1n3e oduro oje1 uorjn rp ur dum tan 103989101Q ur ssooo1d uorjejuoutrioq uorjoung 3ndur ut dumf tu opour uorjequoutro q uorjejuourig yoge g Lpu s 9 uro0 d 17 S Jo ssooo1d uorjequourro qojeq pay uorjeqguourlo YOY pay r1ojo eq jueurquiooo1 Aq uorjonpo d urojo1d 10 1039891 qo1eq po poa 3uopuodop ourrj soo x yoyeq poy Aq sepou adA3 pouou 103 3aye13s Surpoo eurrd Soo x yoyeq poy Aq sjopout d pouou 103 3aye13s Surp feudo IOj 1Od AO UOIDje n2 I2 p9210 T 1039891 S 9AD91 poq P9ZIPM A 9ye3s Apeoys sey ue qod 3soT uomnnos erguouodx q uorjounj snurs Tergu uodxr suomnn os Surseodour K eryuouodxzq P9ZI 8ULIOU ss A poso p UI UOTJ2 9I IOD IO u3 u OTULIOQ3OX SUIT Sep YIM UOI32 91 DIULISYIOXH osop pug jo uoryeor dde ta opour 3uourjreduroo reourT 198 eyep ppou uorjeuoSo1p amp u uoy T 398 eyep popou uorjeuoSo1p amp u ootd z pue T sjos eyep popou uorjeuoSo1p amp u Udo AYA uorjejuoeurlg orxnerp qojeq Dot ouRyIG OBISIAIO 9 A0 uorjequoutrig qojeq poj OURUYH UOL gs Y UOorgoLIgxo eSejsrj nur UMIIQImba snonutquo 10 981 x9 93838 um
322. nally computed 30 a6 4x 515 ODEP4 Bandwidth of Jacobian of right hand side only for im plicit solver must be smaller than NODE In case of ODEP4 0 usage of full matrix is assumed 31 a6 4x 10 4 ODEE1 Final termination accuracy for ODE solver with respect to the relative global error 32 a6 4x 10 4 ODEE2 Final termination accuracy for ODE solver with respect to the absolute global error a6 4x g10 4 ODEE3 Tolerance for solving differential equation initial step size 34 a6 4x 3g20 8 XL X XU Formatted input of NPAR rows each containing three real numbers for lower bound for estimated parameter starting value for estimated parameter upper bound for estimated parameter If the file 7 RES contains the results of a previous run then the corresponding parameter values are read and replace the given ones i e the X values a6 4x i5 SCALE Scale for A A uuas u u uau factors 0 no additional scaling 1 division by square root of sum of squared measure ments 1 division by absolute measurement value 2 division by squared measurement value In case of NCONC gt 0 unformatted input of NTIME NCONC rows for j 1 to NCONC and i 1 to NTIME in this order with the following data t i th measurement time not smaller than zero Cj j th concentration value yp w E measured data i e experimental output and individual weight factor for measurement with number k k 1 NMEAS
323. ndividual area structure In each area a uniform discretization grid is applied Thus different areas with smooth transitions can be used to change the grid size Starting Value for Spatial Interval The spatial interval for the one dimensional space coor dinate is defined by the initial value to be defined here and the individual lengths of the areas defined in the corresponding input table Order of Partial Differential Equation If the right hand side of the partial differential equa tion depends only on first spatial derivatives it is not necessary to evaluate second order approximations for Uy Flux in state equation Flux functions facilitate the input of more complex equations and allow the application of special upwind or similar formulae in case of hyperbolic equations If set to yes there must be variables identifying flux functions and their spatial derivatives in the PCOMP input file moreover a defining equation of the flux for each state variable Spatial Discretization Partial differential equations are discretized with respect to the spatial variable by the following difference formulae 3 point difference formula recursively for second derivatives 5 point difference formula recursively for second derivatives 5 point difference formula for first and second derivatives These difference formulae except the third one can individually be applied to the spatial derivatives of each state variable In case of hyperbolic equat
324. ndmarsh A C 1987 Stiff ODE solvers A review of current and coming attractions Journal of Computational Physics Vol 70 1 62 Caassen N Barber S A 1976 Simulation model for nutrient uptake from soil by a growing plant root system Agronomy Journal Vol 68 961 964 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 Campbel J H 1976 Pyrolysis of subbituminous coal as it relates to in situ gasification Part 1 Gas evalution Report UCRL 52035 Lawrence Livermore Lab Livermore USA Campbell S L Marszalek W 1996 The index of an infinite dimensional implicit system Mathematical Modelling of Systems Vol 1 No 1 1 25 Caracotsios M Stewart W E 1985 Sensitivity analysis of initial value problems with mixed ODE s and algebraic equations Computers and Chemical Engineering Vol 9 359 365 Caracotsios M Stewart W E 1995 Sensitivity analysis of initial boundary value problems with mixed PDE s and algebraic equations Computers and Chemical Engineering Vol 19 1019 1030 Carrasco E F Banga J R 1998 A hybrid method for the optimal control of chemical processes Report Chemical Engineering Lab CSIC University of Vigo Spain Carasso C Raviart P A Serre D eds 1986 Nonlinear Hyperbolic Equations Lecture Notes in Mathematics No 1270 Springer Berlin Carasso C Charrier P Hanouzet B Joly J L 1989 Nonlinear Hyperbolic
325. nfer 2 81 where x JR is the spatial variable with z lt z lt zn and0 lt t lt T Initial and boundary conditions are the same as in the previous section see 2 72 and 2 73 u p x 0 uo p z 2 90 to be satisfied for all x rr R and u p xs E ul p v t D u p LR t ES u p v t 2 91 Ulp mr t M p v t l GE LR t p v t for 0 t T where either Dirichlet or Neumann or any mixed boundary conditions must be defined Also these boundary conditions may depend on the coupled differential and algebraic variables for example when boundary conditions are given in form of ordinary differential equations or in implicit form 51 The right hand side of the partial differential equation depends in addition on the solution of a system of ordinary differential algebraic equations v vg Ua IR given in the form Ov E Gi p u p 1 t u p zi 1 uze p ijt u t gt OVnac Ot Gra p u p Enac ME Enac 1 za D Enges th 254 gt 2 92 0 Gnac 1 Pp u p E EE Tantist Uzs P 5553351 0 1 0 G UD UP Enes t Wa Pies t Ms D s t V t for j 1 n where u p x t is the solution vector of the partial differential algebraic equation Here x are any x coordinate values where the corresponding ordinary differential algebraic equation is to be coupled to the partial one Some of these values may coincide When discretizing the system by the method o
326. nfidence intervals based on some simplifying assumptions The confidence region subject to a given significance level is an ellipsoid which is typically approximated by a surrounding box Whereas small interval lengths can be interpreted as well identifiable parameters larger intervals could be due to degenerated ellipsoids A more rigorous analysis is given in Section 3 1 In many practical situations dynamical models contain too many parameters which are difficult to estimate simultaneously i e are overdetermined An important question is how to detect the relative significance of parameters and how to eliminate redundant ones based on a given experimental design A heuristic approach is presented in Section 3 2 which is 79 computationally attractive and easy to implement The idea is to analyze eigenvalues and eigenvectors of the covariance matrix The absolutely largest coefficient of the eigenvector belonging to the biggest eigenvalue is eliminated and marks a less significant parameter The procedure is repeated until some criteria are satisfied e g reaching a certain significance tolerance The result is a serial order of parameters according to their relevance and which helps to decide which parameters could be eliminated or whether additional experiments should be performed Section 3 2 contains some illustrative examples and a more realistic data fitting problem based on a chemical reaction of an isothermal reactor with too many mod
327. ng Academic Press Orlando Benecke C 1993 Interne numerische Differentiation von gew hnlichen Differentialgle ichungen Diploma Thesis Dept of Mathematics University of Bayreuth Germany Berzins M Dew P M 1991 Algorithm 690 Chebyshev polynomial software for elliptic parabolic systems of PDEs ACM Transacions on Mathematical Software Vol 17 No 2 178 206 Bethe H A Salpeter E E 1977 Quantum Mechanics of One and Two Electron Atoms Plenum Press New York Bettenhausen D 1996 Automatische Struktursuche f r Regler und Strecke Fortschrit tberichte VDI Reihe 8 Nr 474 VDI D sseldorf Betts J T 1997 Experience with a sparse nonlinear programming algorithm in Large Scale Optimization with Applications Part II Optimal Design and Control L T Biegler T F Coleman A R Conn F N Santos eds Springer Berlin Biegler L T Damiano J J Blau G E 1986 Nonlinear parameter estimation a case study comparison AIChE Journal Vol 32 No 1 29 45 42 43 44 45 46 47 48 49 50 Bird H A Milliken G A 1976 Estimable functions in the nonlinear model Communica tions of Statistical Theory and Methods Vol 15 513 540 Bird R B Stewart W E Lightfoot E N 1960 Transport Phenomena John Wiley New York Birk J Liepelt M Schittkowski K Vogel F 1999 Computation of optimal feed rates and operation intervals for turbular reactors Journal of Process Control Vol
328. not allowed to be declared All constants and in particular the structure of the integration interval are defined in the database of EASY FITYVodelDesi n and must coincide with the corresponding numbers of variables and functions respectively Note that initial values for algebraic variables serve only as starting values for applying a nonlinear programming algorithm to compute consistent initial values of the discretized DAE system Example We consider a very simple fourth order partial differential equation obtained from successive differentiation of u x t ae t sin zz Ut QUriyy or equivalently two second order differential algebraic equations Ut 0OUzx 0 U Une 14 defined for O lt x lt 1 andt gt 0 Initial values are u x 0 sin zmz and v x 0 n sin rx and boundary values are u 0 t u 1 t v 0 t v 1 t 0 for all t gt 0 Function u is a possible fitting criterion and a an unknown parameter to be estimated from experimental data The corresponding PCOMP input file is e e a nm C C Problem PDEA4 C Q A o C REAL CONSTANT pi 3 1415926535 C VARIABLE a u V X V X U XX V XX X t C FUNCTION u t ut a v xx C FUNCTION alg equ alg equ v u xx C FUNCTION u_0 u_0 sin pi x C FUNCTION v O v_0 pi 2 sin pi x C FUNCTION uo left u left 0 C FUNCTION u right u right 0 C FUNCTION v left v left 0 C FUNCTION v right v
329. nown experimental data from theoretically predicted values of a model function at certain time values Thus also model parameters that cannot be measured directly can be identified by a least squares fit and analyzed subsequently in a quantitative way In mathematical and somewhat simplified notation we want to solve a least squares problem of the form S 2 pcm min 55 4 h p y p ti t yi 1 1 DSDS Du where A p y t is a fitting function depending on the unknown parameter vector p the time t and the solution y p t of an underlying dynamical system A typical dynamical system is given by differential equations that describe a time dependent process and that depend on the parameter vector p Instead of minimizing the sum of squares we may apply alternative residual norms for example with the goal to minimize the sum of absolute residual values or the maximum of absolute residual values Parameter estimation also called parameter identification nonlinear regression or data fitting is extremely important in all practical situations where a mathematical model and corresponding experimental data are available to analyze the behavior of a dynamical system The main goal of the documentation is to introduce some numerical methods that can be used to compute parameters by a least squares fit in form of a toolbox The mathematical model that is set up by a system analyst has to belong to one of the following categories e explicit
330. nstant alternative values where only the switching points are variable The underlying dynamical system to be considered is now exactly the same considered so far for our data fitting applications It is assumed that either an ordinary differential a differential algebraic equation a one dimensional time dependent partial differential or a partial differential algebraic equation is given It is out of the scope of this software documentation to present a broad introduction into theory and numerical algorithms for optimal control or applications For more detailed information especially also about numerical algorithms and available software in case of ordinary differential equations see e g Goh and Teo 174 Jennings Fisher Teo Goh 236 Bulirsch and Kraft 64 or Machielsen 312 For solving optimal control problems based on distributed parameter systems see Ahmed and Teo 4 Neittaanm ki and Tiba 351 Blatt and Schittkowski 49 or Birk et al 44 Our intention is to consider only control problems that can be solved by the data fitting approach studied so far Thus we suppose for example that we want to minimize a certain quadratic or L3 norm TR Ain s Alp s E u p z t us p 2 t Usalp 2 8 v p t z fi z dv 2 126 by fixing a time value 0 lt t lt T and where integration is performed over all areas confer 2 96 f z is given and our goal is to minimize the distance of a certain criterion f
331. nt piecewise linear or a cubic spline function Given n pairs of real values t1 yi tn Yn we are looking for a nonlinear function interpolating these data In the first case we define a piecewise constant interpolation by 0 t lt ti e t 4 Y stet i 1 n 1 Yn tn St A continuous piecewise liner interpolation function is Yi y d D a L I t Yi Va 44 Sa y onm 141 i Yn y ty St and a cubic spline is given by p t ty t2 ta ta Y1 Yo Y3 Ya t lt t4 s t d A ee grts UI 3 t lt t where p t t1 t2 t3 t4 Y1 Ya Y3 Ya is a cubic polynomial with plti tr ta t3 ta Y1 Y2 Y3 Ya li i 1 shied 4 and Sit ti an Us H Vo Yn a cubic spline function interpolating 71 91 tm Ym subject to the boundary conditions ee e qi hs bss Um Yo Ym Y gt landi m It is essential to understand that the constant and spline interpolation functions are not symmetric Our main interest are dynamical systems say ordinary or partial differen tial equations where the initial value is set to 0 without loss of generality leading to a non symmetric domain Moreover interpolated data are often based on experiments that reach a steady state i e a constant value Thus a zero derivative is chosen at the right end point for spline interpolation to facilitate the input of interpolated steady state data On the other hand any other boundary conditions can be en
332. ntegrator for stiff differential equa tions Computing Vol 26 No 2 355 360 Graf W H 1998 Fluvial Hydraulics John Wiley Chichester Granvilliers L Cruz J Barahona P 2000 Parameter estimation using interval computa tions Report Laboratoire d Informatique Universite de Nantes B P 92208 F 44322 Nantes Cedex 3 France Gray P Scott S K 1990 Chemical Oscillations and Instabilities Clarenden Press Griewank A Corliss G eds 1991 Automatic Differentiation of Algorithms Theory Implementation and Application SIAM Philadelphia Griewank A Juedes D Srinivasan J 1991 ADOL C A package for the automatic dif ferentiation of algorithms written in C C Preprint MCS P180 1190 Mathematics and Computer Science Division Argonne National Laboratory Argonne USA Griewank A 1989 On automatic differentiation in Mathematical Programming Re cent Developments and Applications M Iri K Tanabe eds Kluwer Academic Publishers Dordrecht Boston London 83 107 Groch A G 1990 Autmatic control of laminar flow cooling in continuous and reversing hot strip mills Iron and Steel Engineer 16 20 Gronwall T H 1919 Note on the derivatives with respect to a parameter of the solutions of a system of differential equations Annals of Mathematics Vol 20 292 296 Guckenheimer J Holmes P 1986 Nonlinear Oscillations Dynamical Systems and Bifur cations of Vector Fields Springer New York Gugat M
333. nterval is allowed The number and order of entries must coincide with the number and order of fitting functions of the model function file and of course also with the number of measurement sets given Decimal numbers are rounded to the nearest integer that describes a line of the discretized system Methods for Solving Discretized ODE The parameter estimation code PDEFIT possesses interfaces to several different subroutines for solving ordinary differential equations resulting from the discretization by the method of lines 36 1 Explicit Runge Kutta code DOPRI5 of order 4 5 based on Dormand and Prince formula see Hairer N rsett and Wanner 197 Steplength is adapted internally 2 Implicit Radau type Runge Kutta code RADAU5 Copyright 2004 Ernst Hairer of order 5 for stiff equations see Hairer and Wanner 199 3 Explicit Runge Kutta method of order 4 5 with fixed stepsize for ENO discretization of PDE s to satisfy the CFL stability condition Usage is not recommended in case of non homogeneous equations All codes use dense output i e the integration is performed over the whole interval given by first and last time value and intermediate solution values are interpolated Gradients are obtained by external numerical differentiation Final Absolute Accuracy for Solving Differential Equations Desired final accuracy for the differential equation solver with respect to the absolute global error is to be inserted In case of nu
334. o scale or DAE Model Parameters x Number of Differential E quations E 4 Number of Algebraic Equations Number of Measurement Sets 3 Z Number of Index 2 V ariables alt Lalo lalo Number of Concentration Values 0 E Number of Index 3 Variables A Shooting Index 0 E ple Break Points Consistency Parameters Constraints A A Final Accuracy absolute 1E 10 4 Initial Stepsize 0 0001 x a A Final Accuracy relative 1E 10 4 Bandwidth of Jacobian m Figure 7 14 DAE Parameters the index 2 variables by the stepsize The order of the DAE functions and variables is as follows 1 index 1 variables 2 index 2 variables Number of Index 3 Variables If a higher index system is given define here the number of algebraic variables with index 3 The number is only used for estimating the corresponding error and to scale the index 3 variables by the square of the stepsize The order of the DAE functions and variables is as follows 1 index 1 variables 2 index 2 variables 3 index 3 variables An index i variable i 1 2 3 is defined by the number of differentiations of the variable needed to eliminate the algebraic variables and to formulate an equivalent system of ordinary differential equations Method for Solving Differential Algebraic Equations The code MODFIT can be used for solving parameter estimation problems in differential algebraic equations and executes an 2
335. o the ODE or we use internal numerical differentiation Example 2 5 LKIN_O3 We consider again Example 2 2 now given in the notation Y kiY yi 0 D Yo bam bas y2 0 0 where three different initial doses D 50 D gt 100 and Ds 150 are applied Experi mental data are generated for the same 13 time values starting the simulation from kis 0 1 and ky 0 05 and adding a random error of 5 The differential equation is solved by an explicit integration algorithm with termination accuracy 10 and internal numeri cal differentiation We get exactly the same solution after 26 iterations as for the explicit formulation Only the calculation time is about 50 times bigger because of the extremely ac curate ODE solution where more than 10 correct digits are required If on the other hand we reduce the integration and optimization accuracy to 107 we get the somewhat different solution bus 0 10028 ka 0 04994 again in 26 iterations but now the calculation time is only 18 times bigger 2 5 2 Differential Algebraic Equations Now we add algebraic equations to the system of differential ones 2 31 In this case the fitting criterion h p y p t c z p t c t c depends on ma differentiable variables y p t c and m additional algebraic variables z p t c The dynamical system is given in the form D I wai Fi p y z t c gt y1 0 VW p c Uma Fg PU 2 te tal u e 4 2 36 0 G1 p Y MN gt 2
336. oceed from the following system of m ordinary differential equations where we omit the concentration variable for simplicity y Fi py t 000 p in 2 56 Yn Pain yt um 0 0 gt for 0 lt t lt r yp given initial values j 1 m and H F wt wm s 2 57 m FLOS y t gt l E 8 for Ti lt t lt Titis i 1 ees Mg Tngtl T 31 This formulation is quite similar to 2 40 and 2 41 where switching points are taken into account The difference is the treatment of initial values for restarting the integration In 2 40 and 2 41 these values are known a priori as part of the mathematical model Now the initial values are unknown parameters to be computed in addition to the model parameters p Thus we get mn artificial optimization parameters s1 Sn leading to the total set of optimization parameters Ee det ax ns S gt One has to guarantee that the differences between the trajectories at their right end points coincide with the artificially introduced initial values for the subsequent interval Thus we get an additional set of mn nonlinear equality constraints y p n 81 0 gt 1 y p T2 2 f I 0 7 2 58 y p Tia Sn 0 gt where y p t denotes the solution of 2 56 and y p t the solution of 2 57 for 7 lt t Ti41 t r Tes Example 2 14 LOT_VOL2 A famous biological model describes the behaviour of a preda tor and a prey species of a
337. ocesses Blaisdell Waltham Mass Lapidus L Aiken R C Liu Y A 1973 The occurence and numerical solution of physical and chemical systems having widely varying time constants in Stiff Differential Systems E A Willoughby ed Plenum Press New York 187 200 18 274 275 276 277 278 279 280 281 282 283 284 285 286 287 Lastman G J Wentzell R A Hindmarsh A C 1978 Numerical solution of a bubble cavi tation problem Journal of Computational Physics Vol 28 56 64 Lecar M 1968 Comparison of eleven numerical integrations of the same gravitational 25 body problem Bulletin Astronomique Vol 3 91 Lee J Ramirez W F 1994 Optimal fed batch control of induced foreign protein production by recombinant bacteria AIChE Journal Vol 40 899 907 Lee T T Wang F Y Newell R B 1999 Dynamic modelling and simulation of a complex biological process based on distributed parameter approach AIChE Journal Vol 45 No 10 2245 2268 Lefever R Nicolis G 1971 Chemical instabilities and sustained oscillations Journal of Theoretical Biology Vol 30 267 284 Leis J E Kramer M A 1988 The simultaneous solution and sensitivity analysis of systems described by ordinary differential equations ACM Transactions on Mathematical Software Vol 14 No 1 45 60 Leis J E Kramer M A 1988 Algorithm 658 ODESSA An ordinary differential equation solver with exp
338. odod Som SMOF UL SJUOIJ uorjn ossrp TRISLIO suoryenbo uorjooApe reou po dnoo om T seore uorsngrp 99199 YA 1o3se d VAO o goad snursoo e jo uorgyeSedoud uoryenba uorjooAuo suorgn os snoonbe jo uorjeururejuo uorsngrp pojeururop uorj29Auoo IPON d uorsngrp pojeuruiop uorjooAuoo orporioq umrp ur o3rsoduroo o3rugug soorrjyeur uog e oo OI oseo o1 BNIA umtraqrimb oseud YIM suorjerijuoouoo oseud ping jo uorjeredos juo mo 1ojuno Gerd e13nou oS1euo e jo opour orureu poap qo3ous eq puno4bDsovq o ooo Ch oo oo Oo CC CC CH ooo Ch Ch CH o CH C CQ C5 H 3 C e EH e oo oot oN OQ O C C NAN rd G G c N ri ri ri ri si G CGN CGN CY CH 866 001 96 621 9G GC 09 0 OT OV vr Tv IG 06 GI TZT VG 9 06 OT 08 0 e vi Y vi Cl Cl c zb o G NM DO C Cl TAN e CN ei VISA IVIG SISA IVIG cISATVIC TISATVIC NOISAd TVN YAA OHGAHVG agAL od dAdd MVHUHSINVG oIan AVMA TALO d MOTALO ASO TIVLSAYO ACV 1d SV Id Vdo AJO SOD NINVINOD GAI NOO LAIT NOD WIN dINOO NHDV TIOO TINO LNO SV Id NO 9 Du 49 GS SS SS IS gS gS gS gS GS SS IS gS IS gS gS gS SS 030p ponurjuoo os 96 043 pez eer res eer res GEE DEL 666 S55 See 08T occ Tres O87 fas uorjeor dde eorureu amp pod329 T AYMOUTJWOOSIP YM UOJ DALYRA IOSUO
339. oeurreud ur 3uourLrodxo uorjednjeg uorjoeor uorjeztiourKpod ABMEUTI e uo J I 91 opis puey 1q3t1 ur s nd remue poy uorjenbo erquo1ogtp 1 ss o1 pOT e3our WOI uorgerpeaq SUWE OM Ou JOGO Jo o1juoo ed O gx daynueyy J0q01 e jo suorjenbo 9AOJA Soyei uorjoeor 10j uorjenbo erjuo19grp s uosj1oqoqq WIL orjoqoz xutr 043 Jo o13uoo p euurdo ouit T SOT OAJ YIM DU 30qoqT puno4bDsovq CO Gi lt CH CC Ch CC On Oh Oh Oneu OCH Ch Ch CH Ch OO OO O O O CH 10 O OC st oo Q co Oo Ot CO ed oOo co co EN O O co oo Ga Q 0 Oo E ERAN A PO ARO MOUOS Urso GN TO GN OE OLN ies owe se Foi TG GS GT 0c Y un SET OV 8y 42 IG 0 0 ST ral 6c OV ke CH Ic OT RU 0 Gl 1 N g O MO c QO E CN Go E c Qo i0 NANI CN CO CN QO i0 c c V Ut ADan Is dT IS q NIMS dOd dINIS ODA dINIS THHS d4dHS AUVHS TAUVHS UHMVHS SNS DAISINAS DHSINMWS A VdINWHS UIS ER dOS TOOD HAS dX LVS NOU JS Ind La UATSSHOU oadou LOJON LOgSOH LUWHOS TALO dou INdV dod 9 Du 37 penurjuoo car cav GIG DG gt m gS gS gS gS 0S gS IS gS gOS 0S gS SS 030p oo c N 86 CH CH 10 TITE D e c Edi i GE 9cI Du 6P 67S LOV LCE 667 GLE 931 vez vez 086 vez OST foa uomnnqrassrp axenbs rgo SULIO ogesop ptlos Jo se ogerpourui UOTJNLIASIP eurrou SULI
340. of short peaks the integration routine might not realize the peak at all because of a big time step Moreover the numerical approximation of gradients could become unstable in case of discontinuities Thus MODFIT allows to supply an optional number time values so called break or switching points where the integration of the ODE is restarted with initial tolerances for example with the initially given stepsize The integration in the proceeding interval is stopped at the time value given minus a relative error in the order of the machine precision Note also that break points can be treated as optimization variables i e may vary from one iteration step to the other Copyright 2004 Ernst Hairer Copyright 2004 Ernst Hairer 4 5 Differential Algebraic Equations In this situation EASY FIT Pesis 7 calls again the parameter estimation program MOD FIT as an external executable file where the underlying model is given by a system of differ ential algebraic equations Gradients of the right hand side of the differential equation with respect to system variables are evaluated analytically using either user provided derivatives or automatic differentiation The algebraic differential equation is solved by an implicit Runge Kutta code of Radau type RADAUS confer Hairer and Wanner 199 DAE s with an index up to three can be integrated If consistent initial values cannot be provided by the user the corresponding nonlinear system of
341. of the American Chemical Society Vol 77 3701 3706 Williams J Kalogiratou Z 1993 Least squares and Chebyshev fitting for parameter esti mation in ODE s Advances in Computational Mathematics Vol 1 357 366 Willoughby R A 1974 Stiff Differential Systems Plenum Press New York Widder D V 1941 The Laplace Transform Princeton University Press Winer B J Brown D R Michels K M 1971 Statistical Principles In Experimental Design McGraw Hill Wolf M 1994 Mathematisch physikalische Berechnungs und Simulationsmodelle zur Beschreibung und Entwicklung transdermaler Systeme Habilitationsschrift Mathematisch Naturwissenschaftliche Fakult t Universitat Bonn Wolf H Sauerer B Fasold D Schlesinger V 1994 Computer aided optimization of circular corrugated horns Proceedings of the Progress in Electromagnetics Research Symposium Noordwijk The Netherlands Wolmott P Dewynne J N Howison S D 1993 Option Pricing Mathematical Models and Computation Oxford Financial Press Wouwer A V 1994 Simulation parameter and state estimation techniques for distributed parameter systems with real time application to a multizone furnace Dissertation Faculte Polytechnique de Mons Belgium Yang H Q Przekwas A J 1992 A comparative study of advanced shock capturing schemes applied to Burgers equation Journal of Computational Physics Vol 102 139 159 Yee H C 1985 Construction of a class of symmetric
342. ogrammed in the following way C C BR C C C RI C 100 C C ST SUBROUTINE SYSFUN NP MAXP NO MAXO NF MAXF NR MAXR X Y T C YP YO FIT G DYP DYO DFIT DG IFLAG IMPLICIT DOUBLE PRECISION A H 0 Z DIMENSION X MAXP Y MAXO YO MAXO YP MAXO G MAXR FIT MAXF DYO MAXO MAXP DYP MAXO MAXP MAXO DG MAXR MAXP MAXO DFIT MAXF MAXP MAXO ANCH W R T IFLAG IF IFLAG EQ 0 RETURN GOTO 100 200 300 400 500 600 700 800 900 IFLAG GHT HAND SIDE OF ODE CONTINUE YP 1 Y 1 1 0 X 1 Y 2 X 2 3 X 3 YP 2 Y 2 1 0 X 1 Y 1 X 4 YP 3 Y 3 1 0 X 2 Y 1 T RETURN ARTING VALUES 24 200 CONTINUE YO 1 X 3 YO 2 X 4 YO 3 T RETURN C C FITTING CRITERIA C 300 CONTINUE FIT 1 X 4 Y 2 RETURN e C CONSTRAINTS 400 CONTINUE RETURN G C GRADIENTS OF RIGHT HAND SIDE OF EQUATIONS W R T X AND Y re 500 CONTINUE DYP 1 1 Y 1 Y 2 DYP 1 2 Y 1 Y 3 DYP 1 3 1 0 DYP 1 4 0 0 DYP 1 5 1 0 X 1 Y 2 X 2 Y 3 DYP 1 6 Y 1 X 1 DYP 1 7 Y 1 xX 2 DYP 2 1 Y 2 Y 1 DYP 2 2 0 0 DYP 2 3 0 0 DYP 2 4 1 0 DYP 2 5 Y 2 X 1 DYP 2 6 1 0 X 1 Y 1 DYP 2 7 0 0 DYP 3 1 0 0 DYP 3 2 Y 3 Y 1 DYP 3 3 0 0 DYP 3 4 0 0 DYP 3 5 Y 3 X 2 DYP 3 6 0 0 DYP 3 7 1 0 X 2 Y 1 RETURN G C DUMMY C 600 CONTINUE RETURN C 25 C GRADIENTS OF FITTING CRITERIA W R T X AND Y C 7
343. omposition The tolerance for detecting the rank is set to the same tolerance by which the quadratic programming solver of NLPQLP is called i e to 1072 It must be emphasized that the choice of a tolerance for the approximation of mixed partial derivatives by forward differences has a crucial impact on the performance of the algorithm and should be adapted carefully Internally objective function values are scaled by the starting value i e the initial design 11 Chapter 5 The Modeling Language PCOMP Within the user interface of EASY FIT Vode Design the numerical algorithms are imple mented in a way that the nonlinear model functions defining fitting criteria dynamical model equations and constraints are evaluated either by a user provided Fortran code or by the interpreter PCOMP In the first case one has to code the model function subject to a frame that is inserted in the editor when generating a new problem The usage is completely described by initial comments and is not repeated here In the second case all data variables and functions defining the model functions must be written on a text file in a format similar to Fortran are pre compiled internally before starting the optimization cycle Proceeding from the intermediate code generated function and gradient values are evaluated from the code during run time A particular advantage is that gradients as far as needed are calculated automatically without any numerical a
344. on Stryck O 1995 Numerische L sung optimaler Steuerungsprobleme Fortschritts berichte VDI Reihe 8 Nr 441 VDI Diisseldorf Vossen G Rehbock V Siburian A 2005 Numerical solution methods for singular control with multiple state dependent forms submitted for publication 39 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 Vrar C K 2000 The study of kinetics and mechanism of chlorination of copper I sulphide by chlorine in the presence of oxygen Canadian Metallurgical Quarterly Vol 39 163 174 Vreugdenhil C B Koren B eds 1993 Numerical Methods for Advection Diffusion Prob lems Vieweg Braunschweig Walas S M 1991 Modeling with Differential Equations in Chemical Engineering Butterworth Heinemann Boston Waldron R A 1969 Theory of Guided Electromagnetic Waves Van Nostand Reinhold Company London Walsteijn F H 1993 Essentially non oscillatory ENO schemes in Numerical Methods for Advection Diffusion Problems C B Vreugdenhil B Koren eds Notes on Fluid Mechan ics Vol 45 Vieweg Braunschweig Walter E 1982 Identifiability of State Space Models Lecture Notes in Biomathematics Vol 46 Springer Berlin Walter E Pronzato L 1997 Identification of Parametric Models Springer Paris Milan Barcelone Walter S Lorimer G H Schmid F X 1996 A thermodynamic coupling mechanism for GroEl mediated unfold
345. on model and for computing the parameters in a model Journal of Computational and Applied Math ematics Vol 20 379 385 Stehfest H 1970 Algorithm 368 Numerical inversion of Laplace transforms Communica tions of the ACM Vol 13 47 49 Steinebach G Rentrop P 2000 An adaptive method of lines approach for modelling flow and transport in rivers Preprint No 00 09 IWRMM University of Karlsruhe Steinstrasser I 1994 The organized HaCaT cell culture sheet A model approach to study epidermal peptide drug metabolism Dissertation Pharmaceutical Institute ETH Z rich Stenger F Gustafson S A Keyes B O Reilly M Parker K 1999 ODE IVP PACK via Sinc indefinite integration and Newton s method Numerical Algorithms Vol 20 241 268 Stirbet A D Strasser R J 1996 Numerical solution of the in vivo fluorescence in plants Mathematical Computations and Simulations Vol 42 245 253 Stoer J 1985 Foundations of recursive quadratic programming methods for solving nonlin ear programs in Computational Mathematical Programming K Schittkowski ed NATO ASI Series Series F Computer and Systems Sciences Vol 15 Springer Berlin Stoer J Bulirsch R 1980 Introduction to Numerical Analysis Springer New York Stortelder W J H 1998 Parameter estimation in nonlinear dynamical systems Disser tation National Research Institute for Mathematics and Computer Science University of Amsterdam Strikwerda J C
346. on screen Reports contain the following information e General information about the parameter estimation problem the data and the model as provided by the user e Some numerical problem data for example number of differential equations number of measurement sets etc e User defined tolerances for the parameter estimation algorithm and the subproblem solver e Numerical performance results i e number of function evaluations and final residual e Optimization variables as computed by the algorithm together with starting values and bounds e Parameter estimation data i e time concentration measurement and model values also all weights listed separately for each set of experimental data e Constraint values if restrictions exist In addition we display for each individual set of experimental data a summary of some characterizing statistical parameters DOF degrees of freedom MV mean value of experimental data SOS sum of squares RSOS relative sum of squares square root of SOS divided by DOF MR mean value of residuals absolute values GOF goodness of fit one minus SOS sum of squared differences of MV and model values Especially the goodness of fit value serves as a valuable scaling invariant measure for com paring residuals These data are only displayed if the number of experiments per data set is higher than the number of variables If a simulation run was performed with a positive signifi
347. onal optimization parameters Moreover we have to add equality constraints of the form gi p yi p T c yi p c 0 gt 2 66 Gm p Ymlp T c wL pc 0 to the data fitting problem Here y p t c denotes the solution of 2 65 subject to p and the concentration variable c T could be the final experimental time value i e T l The total set of parameters to be optimized is p p xil oe vs x Example 2 15 CARGO The problem is to transfer containers from a ship to a cargo truck see Teo and Wong 502 or Elnager and Kazemi 137 The dynamical model is described by a system of second order differential equations u t ES T3 i 3 u t and u t describe the driving forces of a hoist and a trolley motor in our case given by third order polynomials with unknown coefficients i e ux t pk pit pit rui k 1 2 The purpose of the original formulation is to generate an optimal control test problem where uilt and us t are to be determined so that a certain cost function the swing at the end of the transfer is to be minimized In our case we suppose that trajectories for the coor dinates x t xa t and x3 t are given between two boundary points a 0 22 0 7 and b 4 22 14 4 0 314 7 The question is how to compute initial velocities v 0 36 exact starting computed solution values solution pi 1 0 1 0 0 7955 pi 5 0 0 0 4 0864 pi 2 0 0 0 3 0344 pl 1 0 0 0 1 3261 p
348. onditions for spatial derivatives follow in the order given by the area data amp p U Ux v t En p U Ux v t Again the function values of u or uz at a suitable end point of an integration area are inserted Moreover r fitting criteria must be defined any functions hi p u Uz Ugg v t h p U us Ure v t u is defined at the corresponding spatial fitting point The final m functions are the constraints gi p Jm p if they exist They may depend on the parameter vector p to be estimated Any other functions are not allowed to be declared All constants and in particular the structure of the integration interval are defined in the database of EASY FIT 4 es9 and must coincide with the corresponding numbers of variables and functions respectively In addition a user may insert further real or integer constants in the function input file according to the guidelines of PCOMP Example We consider a simple example where Fourier s first law for heat conduction leads to the equation Ut Urg defined for 0 lt t lt 0 5 and 0 lt z lt 1 Boundary conditions are at 4L4 0 for0 lt t lt 0 5 and the initial values are u z 0 sin 11 for 0 lt z lt 1 and 0 lt L lt 1 In addition we are interested in the total amount of heat at the surface x 0 given by the equation 2 T T Dems blc vu mm T eL with initial heat k L vg T Function v serves a
349. ons The structure the individual integration areas and the positions of the transition equations are to be defined in form of a table Note that only the status values are obligatory for each individual partial differential equation It is sufficient to define the corresponding information only within one line that identifies the area In this case the equation number must be 1 For each area and the differential equation identified by a serial number the following data must be set 42 Name The area may be characterized by an arbitrary string Size The length of the spatial interval of the corresponding area must be given Lines Within each area an equidistant grid with respect to the spatial variable is used to transform the PDE into a system of ordinary equations The number of grid points must be odd to get an even number of equidistant intervals for applying Simpson s rule in case of numerical integration with respect to the spatial variable Note that the larger the number of grid points is the larger is the result ing ODE Status A status number identifies the type of the transition condition be tween areas and also of the boundary conditions for each individual equation The following options are allowed for the left and sepa rately for the right end point of the area 0 no boundary condition 1 Dirichlet type boundary condition function value given 2 Neumann type boundary condition derivative value given 3
350. ons f p depend on the solution of a dynamical system Moreover we describe a couple of extensions of the data fitting problem and the dynamical system to be able to treat also more complex practi cal models Most examples contain the name of the corresponding test problem of the EASY FIT Pess database from where implementation details and further data can be retrieved in some cases only in modified form 2 1 Introduction The basic mathematical model is the least squares problem to minimize a sum of squares of nonlinear functions of the form T 2 min 1 fi p pe BC 2 1 PLS p lt pu Here we assume that the parameter vector p is n dimensional and that all nonlinear functions are continuously differentiable with respect to p Upper and lower bounds are included to restrict the search area f p is a suitable fitting criterion which may depend on the solution of an underlying dynamical system e g a system of ordinary differential equations Alternatively the L2 norm may be changed to another one e g to minimize the maxi mum distance of experimental data from a model function Thus we formulate either the L4 problem min xxu fi p PLS p lt Pu or the L4 problem min max 1 fi p pe R 2 3 Pi lt p lt Pu i However we assume that our models a dynamic i e depend on an additional parameter in most cases the time In addition there might be an additional independent model parameter by which e g a concent
351. ons y p c y depending on the parameters to be estimated the actual concentration value and the solution of the previous interval at the break point 7 Internally the integration of the differential equation is restarted at a switching point Example 2 7 LKIN BR Consider the linear compartment model of Example 2 2 Y kpy 41 0 Do 1 0 Do ge yo bam kan y2 0 0 17 Linear Compartment Model with Switching Points 100 90 80 70 60 y p t 50 40 30 20 10 Figure 2 5 Function and Data Plot for Compartment 1 with an initial dose Dg 100 for the input compartment y After 7 24 respectively T9 48 time units another dose of D 40 respectively Dy 40 is applied Formally the initial values at t 0 and the switching times are given by ie Do yp 0 vilo yi o 71 yip T D T P yalp T1 w9 mm vilo yi p 72 yy p 72 D Jal Y2 P T2 Y2 P T2 Here we omit the concentration variable c to simplify the notation and the parameter vector is p Kio ka Experimental data are simulated for 17 time values between 0 and 100 and ki 0 1 and ky 0 05 A random error of 5 as added to the obtained data The differential equation is integrated by an explicit Runge Kutta code with absolute and relative termination accuracy 1077 The least squares solver DFNLP is started at kia 1 and ko
352. ontains last minute changes a summary of new features and es pecially the information how to transfer parameter estimation problems from easier versions of EASY FIT Pesisn to the new one Welcome Form Important FASQV FEI ModelDesign DES ML is an interactive software system to identify parameters in systems of explicit model functions dynamical equations steady state systems Copyright Prof K Schittkowski SC Laplace equations i Dept of Computer Science Ordinary differential equations stiff non stiff University of Bayreuth differential algebraic equations D 95440 Bayreuth one dimensional time dependent partial differential equations one dimensional partial differential algebraic equations Synonyms Parameter estimation data fitting experimental design mathematical modelling simulation 1 Usage is restricted according to the valid license agreement 2 Select favorite editor and graphics system in subsequent form or by Utility menu item System Configuration 3 Check the compiler and linker calling sequences before generating own Fortran code 4 You are NOT allowed to distribute this software to any other person or organization 5 If the system reacts too slow on your computer please delete subset of demo problems you do not need do not accept the license conditions Show license conditions Figure 2 Welcome Form If something goes wrong please cont
353. order of the above program blocks They may be repeated whenever desirable Data must be defined before their usage in a subsequent block All lines after the final END statement are ignored by PCOMP The statements within the program blocks are very similar to usual Fortran notation and must satisfy the following guidelines Constant data For defining real numbers either in analytical expressions or within the special constant data definition block the usual Fortran convention can be used In particular the F E or D format is allowed Identifier names Names of identifiers for variables and functions index sets and constant data must begin with a letter and the number of characters i e letters digits and underscores must not exceed 20 Index sets Index sets are required for the SUM and PROD expressions and for defining in dexed data variables and functions They can be defined in different ways 1 Range of indices indi 1 27 2 Set of indices ind2 3 1 17 27 20 3 Computed index sets ind3 5 i 100 i 1 n 4 Parameterized index sets ind4 n m Assignment statements As in Fortran assignment statements are used to assign a nu merical value to an identifier which may be either the name of the function that is to be defined or of an auxiliary variable that is used in subsequent expressions ri pl p4 p2 p4 p3 p2 11 r2 pl 10 p2 p3 p4 p2 p4 p3 pl f rl 2 r2 2 Analytical
354. orf Majer C Marquardt W Gilles E D 1995 Reinitialization of DAE s after discontinouities Proceedings of the Fifth European Symposium on Conputer Aided Process Engineering 507 512 Mannshardt R 1978 One step methods of any order for ordinary differential equations with discontinuous right hand sides Numerische Mathematik Vol 31 131 152 Maria G 1989 An adaptive strategy for solving kinetic model concomitant estimation reduction problems Canadian Journal of Chemical Engineering Vol 67 825 837 21 319 320 321 322 323 324 325 326 327 328 329 330 331 Marion M Mollard A 1999 A multilevel characteristics method for periodic convection dominated diffusion problems Numerical Methods for Partial Differential Equations Vol 16 No 1 107 132 Marquardt D 1963 An algorithm for least squares estimation of nonlinear parameters SIAM Journal of Applied Mathematics Vol 11 431 441 Marquina A Donat R 1993 Capturing shock reflections A nonlinear local characteristic approach UCLA CAM Report No 93 31 Dept of Mathematics University of California at Los Angeles Marquina A Osher S 2000 Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal Report Dept of Mathematics University of California Los Angeles Martinson W S Barton P I 1996 A differentiation index for partial
355. ormed numerically Gradients can be evaluated by automatic differentiation without additional round off truncation or approximation errors and without compiling and linking of code Ordinary differential equations may become stiff and large We introduce explicit and implicit methods and exploit band structures Parameter estimation problems based on unstable differential equations can be solved by the shooting method Various types of one dimensional partial differential equations are permitted also hy perbolic ones describing shock waves Advection diffusion transport or related equa tions can be solved successfully by non oscillatory discretization schemes even with non continuous initial or boundary conditions Partial differential equations may be defined with Neumann and Dirichlet boundary or transitions conditions Moreover these conditions can be formulated in terms of algebraic equations coupled at arbitrary spatial positions Algebraic partial differential equations may be added to the time dependent ones Data can be fitted with respect to the Ls the L or the L norm i e with respect to sum of squares sum of absolute values or maximum of absolute values of the residuals 17 A statistical analysis provides confidence intervals for parameters depending on an user provided estimate for the variance Moreover correlation coefficients and covariance matrix are computed 18 Proceeding from the inverse of the
356. os amp repunoq orporrod yya suoryenbs uorgosape po dnoo oA G u uorjooApe Kpeojsun reoumquoN Z U uorjooApe Kpeojsun reoumquoN juoarorgooo UIT o qerreA YM suoryenbo uorooApy juerorgooo o qerreA yy uorgenbo uorooApy suorjenba Ajratyisuos yyim Arepunoq yo urog osem uorenbo payeuruop uomooApy Arepunoq 3jo Wor osem uorjyenbo pojeuruiop uonooApy gyep erur uueurong q3r4 uorjenbo uorsngrp uorjooApy SYSIp uorjoriooe uerio dow ur soan urmnriqmmbo euni T UOT ISUBI po opour Apyrordxo TILA soue1quioutr oA T GO perdnoo pue seore om ur ALT Sor uorsngt q seore oA ut MEJ sor uorsngt q punoAbDso2v0q 0 0 GE 0 UG 09 0 UG 06I 0 UG 06I 0 UG S i 0 0 I O8T 0 0 I 8cI 0 0 I TZT 0 0 I S 0 I I 0G 0 0 I GE 0 c 6 06 0 0 I 0c 0 0 I 0G 0 0 Op 0 0 I 09 0 0 Sv 0 0 T OST 0 0 OSI 0 0 OF 0 I 9 0 I I TOT 0 0 I VG CN NN ON e o Q 0 CH MOTH UV NIdJJV VGLOWAGV GLOWAGV S LOHAUV LAV N LOUACV LOWAGV SOHAUV dd OTAGV QNTOFACV dO OWMAGV NS DHAGOV NG OMAGV OLA ACV OA ACV SNOG ACV WOCd ACV AHI ACV LA OM NV4gIN3ING O SVAUVG SVAUVG IWDU suoryenbg teryu r gtq Tened 797g 21001 47 gS gS gS gS IS gS gS gS gS gS 0S 05 SS 030p penurjuoo 9sz GET oct O87 918 loss ECH OCP ECH OCP O87 vVE rea Jor eer rer 661 09 Fc GGT 971 suorjoeor reoutpuou YM uorgenbo y 10dsue1 DAISIOASIP IALJDIAUO
357. ouromseour re norguo odeys suo ortoydsy sjuouromnseour re norguo odeys suo otiroudsy sjuouromseour re norguo odeys suo ortgudsv eyep jo uorgeurrxoadde euorjey SO QBLIBA UI SIO LIO pue uorj eor I9plo 3SIY qISISA 11II YM Y LSO neqerpe 9o3e3s Apeo3g osexur req moj e jo ustse q So qerreA popou quopuodopur 99113 YYLM uro qo1id ourop y eurrurur BOOT 043 YPM uro qo 1Id 3897 oruropeoy S193oure red pont quopuodopur g tu uro qo1q puno b390q CO CC CC Ch CH oo Ch Ch Ch Ch CH N N oo Oh Ch Ch CO Ch mu CO NA NA N OO O un 0 0 au tu VIG 69 I 9 0G VI val VG S S LG S OT 0G VG 209 OT l G El o cO do cO cO x c0 c Co N 1 elt a o TLAYIMHO VLSONHHO SOSNAO LNSHINHO dus LVO AA HHOUnd qogaxod 88DDOd CSODOT THING SLLANNGE XM dOULV V HM4HdSV UHHASV UAHdSV TUHHASV EXUAAV ULSO IV ANT UVdP SAVACONE SAHTIVAZ SUVACNIG IWDU uom unid PONT WAXY T A 21091 ponurjuoo gS gS a a gS a a a 0S a a a gS 0S uou a DI DD DD DG m 9uou 030p rc TOG VVG TVG CHL WI Sutj3g ejep erguouodx q uors1odsrp pue Suruodoojs reout uou yya vrpour ur uoryeSedodd OAR uoryeurrxoadde peuorje1 uorjoeod ourAzu q Xpn3s LSIN soouoJogrp rnss d otrequdsouny uoyenbs JACA equa oseo p l U pinbr e ur Surjegedo1d sore Xpn3s LSIN 9 uej33rusuer 92U919L193UI TEMNO Sumu ejep erguouod
358. out the boundary location The vari able is TRUE if and only if the actual X value defines the left boundary Logical to inform the user about the boundary location The vari able TRUE if and only if the actual X value defines the right boundary Flag defining the desired type of calculation 0 Execution of SYSFUN before requiring function or gradient values e g for preparing common s 1 Evaluate flux functions of PDE s and store results in FLUX 2 Evaluate right hand side of PDE s and store results in UP 3 Evaluate initial values of PDE s and store results in UO 4 Evaluate right hand side of coupled ODE s followed by coupled algebraic equations and store results in VP 5 Evaluate initial values of coupled ODE s followed by initial values for algebraic equations and store results in VO 6 Evaluate transition functions for PDE s 7 Evaluate transition derivatives for PDE s 39 8 Evaluate fitting criteria and store results in FIT 9 Evaluate constraints and store values in G 11 Evaluate the partial derivatives of flux functions subject to U UX and X and store results in FLUXU FLUXUX and FLUXPX FLUX MAXPDE When calling SYSFUN FLUX contains the cofficients of the flux functions of the partial differential equations FLUXX MAXPDE When calling SYSFUN FLUXX contains the coefficients of the first derivatives of the flux functions FLUXU MAXPDE Has to get the derivatives of the flux functions for
359. oy for the external graphics system GNUPLOT The standard plot program is implemented in form of a separate Fortran program called SP_PLOT EXE Plot data generated by MODFIT or PDEFIT are passed directly to SP_PLOT and GNUPLOT on files and are not kept in the database A user has the option to require also an overlay of function and data plots Three dimensional plots can be viewed from different angles Copyright 1986 1993 1998 2004 Thomas Williams Colin Kelley Plots can be generated for the public domain software GNUPLOT that must reside in a directory The corresponding command file to start the program is called GNUPLOT GNU Optionally this file can be edited before starting GNUPLOT to modify and adapt plot commands It is possible to change the display style or to start a printout To get the corresponding pop up menu one has to press the right mouse bottom It is very important to close the plot correctly by answering the GNUPLOT request correctly Otherwise the system may break down when trying to get the next plot Plot data are read in directly from a text file For problems with at most one concentration value or at most one time value two dimensional plots are generated that show the experimental data and the model function values within the interval determined by the first and last time or concentration value respectively In the first case measurement and model values are displayed over time in the second case over concen
360. p pi7 t sin zz is the exact solution in case of p py pa 1 1 T 44 Heat Equation Figure 2 24 Solution of a Parabolic Equation To construct a data fitting problem we simulate experimental data for p 1 1 at 9 time values t 0 1 tg 0 9 and 3 spatial values x 0 25 23 0 5 and zs 0 75 Subsequently a uniformly distributed error of 5 is added Thus the data fitting problem consists of minimizing the function 3 2 k 1 u p Tk t mi ur d 9 over all p IR The partial differential equation is discretized by 31 lines and a fifth order difference formula The resulting system of 29 ordinary differential equations is solved by an implicit Runge Kutta method with integration accuracy 1079 When starting the least squares algorithm DFNLP of Schittkowski 29 with termination accuracy 10779 from pg 2 2 we get the solution p 0 98 0 97 after 9 iterations The final surface plot of the state variable u x t is shown in Figure 2 24 2 6 2 Partial Differential Algebraic Equations One dimensional partial differential algebraic equations PDAE are based on the same model structure as one dimensional time dependent partial differential equations The only difference is that additional algebraic equations are permitted as in case of DAE s Typical examples are higher order partial differential equations for example Ut Jf p U Ucarra T t gt 45 or distributed sys
361. p 20 j mtl Mr pi lt p lt Pa where v p t is the solution vector of an optional system of n coupled ordinary differential and algebraic equations similar to the previous section It is assumed that all model functions hj p U Ux Ure v t and g p are continuously differentiable subject to p for k 1 r and j 1 m and also the state variables and their spatial derivatives u p x t u p v t Uss p x t and v p t Test problems with one dimensional partial differential algebraic equations are listed in Table B 7 Not listed are the number of integration areas switching times and structure of the boundary conditions There are no equality constraints 61 Dono gS SS IS SS gS gS gS IS IS gS gS TS OS SS SS SS GS 030p 107 los Ir 688 Loc oce Los lozs po uorsi d stp pue Suru d 1s 1eouruou yy erpata ur uoryeSedodd SARA suoryenba ore1qoSTe qr uorjeor dde orureu amp pod399 d uoryenbo s10S ngp SoLtA 9P 39M9910M oqn3 orjsepo pog pr bi e ur SuryeSedold sore M oudoijseqeo dsno tw os nduit oA10u p ouso u I IG o quourrod ruros yya m xt pmby amp reurq JO SMO 3uo Lmo 193unoo o qrsso iduroout OM T OLT 3q uo uorepKxo OO umriqimbo oseud tu suorjerjuoouoo oseud ping Jo uoryeredos 3uo Lmo 1o03unio SUSISOI orgdeiSojeuro1euo 10 329go 310dsuvi OS TO IJP JopUN 1o3 eA YIM Dot edeo Uorjn os DAOU YIIM wo qoid onTeA Arepunog Pls uorje
362. p is also called the ordinary least squares estimator OLS to distinguish it from alternative techniques for example from the weighted or generalized least squares estimators The question we are interested in is how far away f is from the true parameter p It is assumed that the independent model values t are given a priori without errors and that e denotes the statistical error of the measurements or the response variable respec tively As usual we suppose that the errors e y h p t are independent and normally distributed with mean value zero and known constant variance o2 i e e N 0 0 for E The basic idea is to linearize the nonlinear model in a neighborhood of p and to apply linear regression analysis since linear models are very well understood see Seber 456 By defining f p h p ti bin ti 8l and e y f p q p p e e1 4 7 y yy Y1 we get from the first order Taylor expansion s p lf vl lLf p VF py o y IVA ell Here denotes the Euclidian norm We denote by F V f p the Jacobian matrix of f p at p p and assume that F has full rank A solution of the linear least squares problem is immediately obtained from the normal equations d o Co Fre Q from which we get a first order approximation of the solution p by p p dk ER E From this approximation some statistical properties known for linear models ca
363. parameter vectors are pi pi ana pt i 1 2 3 Moreover we determine s s p l n an estimate of the variance o Parameter pt can be estimated for all sample sizes quite successfully The variance estimates converge to the true value as expected Figure 3 1 shows the fitted data for l 120 measurements 3 2 Significance Levels by Eigenvalue vector Analysis of the Fisher Information Matrix Proceeding from a parameter estimation model corresponding data and a successful least squares fit significance levels of the estimated parameters are to be evaluated If a model seems to be overdetermined i e contains too many parameters compared to the number of equations the levels give an impression of the significance of parameters and help to decide upon questions like 83 Table 3 1 Confidence Intervals for Different Sample Sizes l 120 60 i123 1 15 o 0 0998 0 0963 0 0945 0 0891 p 0 0998 0 0996 0 0989 0 0973 pi 0 1025 0 1032 0 1036 0 1055 D 0 942 0 918 0 889 0 281 Do 0 986 0 997 1 042 0 977 Pa 1 030 1 076 1 196 1 673 p 93 7 91 4 87 6 22 7 Ds 98 5 100 1 104 2 98 6 which parameters can be identified which parameters can be treated as constants whether additional experimental should be added or not Moreover overdetermined data fitting problems lead to unstable and slow convergence of Gauss Newton type least squares algorithms with a large number of iterations until termi nation tolerances are satisfied
364. pect to the relative global error a6 4x g10 4 ODEE2 Final termination accuracy for ODE solver with respect to the absolute global error In case of TVDRK the pa rameter is used to pass a factor for reducing the stepsize if the CFL condition is not satisfied gt 1 in this case El a6 4x g10 4 ODEE3 Tolerance for solving differential equation initial step size 34 lis a6 4x g10 4 XSTART Value of the 9 component at the leftmost bound a6 4x 810 4 Formatted input of NCPB 2 NPDE 1 lines 5i5 Each headline contains the name of the area the spatial size of the area the number of discretization points in the area 10x 315 whereas the following NPDE lines contain for each PDE status of left boundary condition status of right boundary condition spatial derivative approximation APRMET 6 The boundary status is 0 no boundary condition 1 Dirichlet boundary condition 2 Neumann boundary condition Possible spatial derivative approximations are 0 order taken from APRMET 1 central differences for uy and recursive application for uz DQUPOI gt 2 2 5 point differences for uy and recursive application for u DQUPOI gt 4 4 Forward differences for uz 35 5 Backward differences for uy a6 4x 3820 8 Formatted input of NPAR lines each containing three real numbers for lower bound for estimated parameter starting value for estimated parameter
365. pplications Part II Optimal Design and Control IMA Volumes in Mathematics and its Applications Volume 93 271 300 Springer Verlag Berlin Heidelberg and New York Godfrey K R DiStefano J J 1985 Identifiability of model parameters in IFAC Identifi cation and System Parameter Estimation P Joung ed Pergamon Press Oxford 89 114 Goh C J Teo K L 1988 Control parametrization A unified approach to optimal control problems with general constraints Automatica Vol 24 3 18 Gonzales Concepcion C Pestano Gabino C 1999 Approrimated solutions in rational form for systems of differential equations Numerical Algorithms Vol 21 185 203 Goodman M R 1974 Study Notes in System Dynamics Wright Allen Press Cambridge MA Goodson R E Polis M P 1978 Identification of parameters in distributed systems in Distributed Parameter Systems W H Ray D G Lainiotis eds Marcel Dekker New York Basel 47 134 12 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 Goodwin G C Payne R L 1977 Dynamic System Identification Experiment Design and Data Analysis Academic Press New York Gorfine M Freedman L Shahaf G Mehr R 2003 Maximum likelihood ratio test in complex models An application to B lymphocyte development Bulletin of Mathematical Biology Vol 65 1131 1139 Gottwald B A Wanner G 1981 A reliable Rosenbrock i
366. pproximation errors 5 1 Automatic Differentiation Let f p be a nonlinear differentiable function with real values defined for all p JR By automatic differentiation we understand the numerical computation of a derivative value V f p of f at a given point p without truncation errors and without hand coded formulas Numerical differentiation requires at least n additional function evaluations for one gra dient calculation and induces truncation errors Although very easy to implement the numerical errors are often not tolerable especially when the derivatives are used within an other numerical approximation scheme A typical example is the differentiation of solutions of unstable differential equations in a parameter estimation problem Automatic differentiation overcomes the drawbacks mentioned and is a very useful tool in all practical applications that require derivatives The resulting code can be used for the evaluation of nonlinear function values by interpreting symbolic function input without extra compilation and linking Whenever needed gradients can be evaluated exactly at run time There exists meanwhile a large variety of different computer codes for automatic dif ferentiation see Juedes 243 for a review They differ in the underlying design strategy domain of application mathematical method implementation and numerical performance The code PCOMP that is to be introduced now is based on a somewhat restricted language
367. q Sour Zuo DAIJBIDOSSIP Jo uonrqrqu o1njxrur IoyBM pue oue3nqost 10 o1n3e1oduro mod m q 9T19Y8UL OTJSOTO ODSTA JO uorjeuriojgo q suorgnjos ourure snoonbe ur Kjyr rqnjos ZOD 103 opout HTA jejsoulroqo e JO aut oouoprso1 feudo urojs s umnriqr mbo eoruroq Hd Jo uorjoung se perjyuo3od o19z Jopour uorje npo1 oS1eu uorjeuruLiojop JYSIOM re noo our 10 oSnjrrijuooe n uorjerjuoouoo Togo poo q uorjexo duroo 99ejms yya uorndaospy punosbyanq N NA cO co SH St SIE el t Hem C I ot eh SENE 2907 E ENC EN 9T VG GG 0c ve OT e I 0 0c GI 0 Ic LET VI OT 0c TT GE OT YN He P la Go o AA Gi Dr i Cl r Cl AN P CO kO a STOT IOWSH VIDITONY ETOTIO TY GLOT IOWSH TOTIO HTA OTOTIO TY TIOIIOWMH GIOTTO AY VIudd YWLSO VN ISO LINN ANVHLAN IMOT LUN MOT UN ZNT SSId LNIOdM4QA IN HO ITOS ZOO LV LSINGHO nOW INTHO ADUVHO TULNAO S doo ld duOsdv IWDU suoryenbg oyejg peojg g NWI 16 10 on MO c ri 10 n an m on DD DG GIG D 030p 80 STI STI STI STI STI STI STI STI STI STI STI 6q1 6q1 NOT GST LOV LOV LOV LOV LOV LOY LOV LOV LOV LOV LOV soouejs qns OA JO uorjeuruLijop JYSTOM re noo our 10J srsA eue gyep en oouejs qns ouo 10 uoryeururiojop JYSTOM Iepnoo our 10j sIsATeUR e3ep ISNFLAJUIICIAN oururejqg3oourure Au q39UItp N N JO uorjexr ormouromuo30q DOUT inqd ns 03 oprxorp ud
368. quations with initial ODE 5 Edda values concentration values and switching points system of differential algebraic equations up to index 3 with initial values concentration values and switching points system of one dimensional time dependent partial differential equations with initial and boundary values disjoint integration areas coupled ODE s D Parameters to be Estimated Update Simulation Data Fitting Experimental Design 1 0000E 00 1 0000E 00 ODE 05 1 00E 05 6 0377E 00 2 flux functions and switching points same as for PDE but with additional algebraic FUE partial differential equations and coupled DAE s EI 1 00 05 0 00E 00 1 0000E 02 1 0000E 00 100E 05 37099E402 1 Model Parameters Language PCOMP FORTRAN Nom Sum of Absolute Residual Values Sum of Squared Residuals Maximum of Residual Values 3 Model Functions m Record nl 4 I 1173 E ou p of 1398 Model name optional Figure 7 1 Main Form 7 1 2 Documentation Text There are a few information strings that are useful to identify different parameter estimation problems particularly if a sequence of experimental data sets is to be processed These strings are inserted in the reports generated by EASY FIT 9 and the plot output Note that the proposed meaning of the items has no additional impact on th
369. r ns jo uorjeprx UOI32 91 93e3s Ape 1 uoryenba o gutg 109d9991 uo puesi pjoo ALM oA1no quouroove dst T pueSt orpe1 ouo pue s109d9991 OM OAIN uorjeanjeg 109d9991 wo punoduioo ne Jo sa mo quourooe dst T oAIno quourooe dst T spues om pue 103dooor ouo YIM opour umnriqrmbo sse spurs pue sio3dooo1 omy ooueysqns YIM punoduroo H e jo aamo juourooe dst T pueSrporpex ouo pue s103dooo1 OM oAIND uorjemnjeg spues omg 103dooo1 ouo VOI punoduroo He Jo samo quourooe dst q spues omy 103dooo1 ouo yy opour urnriqr mbo sse pues ouo pue 103dooo1 ouo YALA opour umnriqrmbo sse pues ouo pue s1o3dooo1 om YIM opour umriqrmbo sse puno4bDsovq NM CO C cO bh vc OM i ge e e e e wf TS 06 0 GG OT vr GI IT GL OT LG VI 0 Y MASON X c c e NA s mot cVULIn IVULIN OLLVULLL YUNHITAS ovu SS NOT ONIS GOU SOU LDTIOSTNH 9DITOAY SOITTOA VOTIO TY SITIO HAY GOI IOWH STOTIO TY IOTIOHY 9TOTIO TY IWDU Tf 10 4 Ordinary Differential Equations As before we proceed from r data sets of the form iu E Sieg s S where I time values le concentration values and l ller corresponding measurement values are given Furthermore we assume that weights we are defined However some of the weights UE can become zero in cases when the corresponding measurement value is missing if artificial data are needed or if plots are to be generated for state variables for which E
370. rameter Values no algebraic equations and two ordinary differential equations coupled at boundary points tr lend te 0 1 CL Dalelp az t z p 9i t esp z r t gt Da c p zn t 2 p 55 t e p En t CR Here we have ale z exp d 1 0 Initial values are elp 2 0 0 ps 0 0 cz p 0 0 and cg p 0 co The coupled ordinary differential equations are inserted to model the flux into and out of the system Thus the boundary values are c p xr t cr p t and ein xr t cr p t and the fitting criteria are the two functions h p t k cr p t cn p y with a scaling constant k 100 000 The remaining coefficients are supposed to be parame ters to be estimated i e p D b q co Experimental data are simulated at 10 equidistant time values between 0 and 5 subject to a 5 error Exact starting and final parameter values are given in Table 2 11 They are obtained by executing DFNLP with a termination tolerance of 107 terminating after 56 iterations The partial differential equation is discretized at 21 lines and a three point difference formula The resulting system of ODE s is solved by an implicit method with an error tolerance 1075 Function and data plot is found in Figure 2 28 and the state variable c is displayed in Figure 2 29 If coupled algebraic equations violate initial conditions 2 93 after a suitable discretiza tion of the partial derivatives u and Urz Newton s method
371. rameter identification and model verification in systems of par tial differential equations applied to transdermal drug delivery to appear Mathematics and Computers in Simulation Schittkowski K 2008 NLPINF A Fortran implementation of an SQP algorithm for mazimum norm optimization problems user s guide version 2 0 Report Department of Computer Science University of Bayreuth Dai Y H Schittkowski K 2008 A sequential quadratic programming algorithm with non monotone line search Pacific Journal of Optimization Vol 4 335 351 Schittkowski T Br ggemann Mewes B 2002 LII and Raman measurements in sooting methane and ethylene flames Report LTTT Dept of Applied Natural Sciences University of Bayreuth Schneider R Posten C Munack A 1992 Application of linear balance equations in an online observation system for fermentation processes Proceedings of the IFAC Modelling and Control of Biotechnical Processes Boulder Colorado 319 322 Schreiner T 1995 Mechanistische und kinetische Parameter der Arzneistoffaufl sung aus festen Zubereitungen als Kriterien der galenischen Qualitdtssicherung Dissertation Dept of Pharmaceutics University of Saarbr cken Schumacher E 1997 Chemische Reaktionskinetik Script Dept of Chemistry University of Bern Switzerland Schwartz A L 1996 Theory and implementation of numerical methods based on Runge Kutta integration for solving optimal control problems Ph D Th
372. raqimb o durtg SCHO YIM 10998 19X0 8 1sr n puno4Dsovq ooo ooo CH o Oc c CC CC cn On Ch CC ooo ooo CH ooo ooo CH OO YN i O C 1D o N CO CC CC CC CC oOo CH Ch Ch Ch Ch ooo CH NO TO co oo CN hi CN c0 c CGN CN OO Go CH 001 OST 9cI 9G c6T O8T 08 OT OT gy 8 Gs 09 001 ST 9 IcVI UC js 69 001 VI 0G ST MIO ei Y ch 2 MN CN c co r oO ti t n ai CO Gi CGN CGN CGN x cO cO eh dNALNYAA JOYANYAAA LNINAHU LENANYTAA HO LVHGH4 LVg8GSJ4 OTCHA HLVH WA LV8 dad dVAG O4 Uda LSVA TOS dXM NIS dXM ONT dXM NYITHLOXA OVA OXA MV dd XH C TIAH LA TIAHLA TAHLA NYIAHAH LA TONVHL4 VO YUALSA ILIANO4A XHOMH MOVIDA IWDU a ponurjuoo q q q IS q q IS os Turi q q q eS A a ol q al q q q gel q gz q gz q sce q gz q q q q q A 020p STE 98 T9F 0 S 491 P amp Z P amp Z e67 80 80 80 Tr6 TIF TIP poe poe poe poe poe rr rr FT Peg SOLJOUL unsur pue so nIS3 103 opour erurur uorjoeoi soon gt o1ogdsourje pue sjuourjuoo UDIMI oq uoqieo pue 19jeA4 KS1ouo Jo AJUBIXA TIPO ZOD eqo 5 putmdn Ota dopi g Jo 30311 uoryeguouri 9784S pigos ur uorjonpoad ey pue q3m013 romyrfny e 9 19Q 13 Sut opo A xu polis Y ur 19jsuer sseur pue Surxrur pmbry sery xu polis Y ur 19jsuer sseur pue Su
373. ration or temperature value is to be specified from where a set of measurements is obtained To illustrate the situation we omit possible additional data sets dependencies on un derlying dynamical systems and constraints on the parameters and differ between three situations 1 Time dependent models The model function f p depends on the experimental time i e we have measurements of the form ti Yi i l l 2 4 moreover a model function bin t and we want to estimate the parameter vector p by minimizing S 2 min X h p ti yi me for 2 5 pi lt p lt Pu In this case we define 2 Time and concentration dependent fitting criteria The data fitting function f p de pends on the experimental time and an additional parameter which we call concentra tion Any other physical meaning is of course allowed We proceed from measure ments of the form aw i lopez 2 7 a model function h p t c and we want to estimate the parameter vector p by mini mizing SM 2 min 2 1 h p ti ci yi pe f m dw w 2 8 pi lt p S Pu In this case we define Advantages are the possibilities to define a model as a function of t and c and to generate three dimensional plots The drawback of this formulation however is that an underlying differential equation cannot depend on c as well since we would have to evaluate the right hand side of an equation also at intermediate times and would not know how
374. re or estimate the volume and the structure of the ellipsoid The most popular ones are D det C p q A trace C p q A trace 1 p q d E Amin I p q E or C Amnd p Amas ll p q Here Ao Il p q and Amar 1 p q denote the minimum and maximum eigenvalues of I p q For a more detailed discussion see e g Winer Brown and Michels 551 or Ryan 414 For our numerical implementation we use the A criterion since the computationally attractive confidence intervals by which the size of the ellipsoid is estimated take only the diagonal elements of the covariance matrix into account see 3 6 This leads for each p IR to the optimization problem min trace C p q gj p q j 1 Me q e Im fp d 3 15 g p q O ges vss C gy where we add additional bounds for the variables q and additional equality and inequality constraints depending on the given model parameters p and the design variables q to be computed There remains the question how to compute the derivatives of the objective function q trace C p q 91 subject to q in an efficient way Numerical differentiation of q subject to q by a difference formula based on a previous numerical differentiation of h p q t subject to p by another difference formula is unstable because of accumulation of truncation errors It is assumed that second order analytical partial or mixed partial derivatives are not av
375. re satisfied see Figure 2 38 69 Unconstrained Heat Equation t p ives a ivat Second Partial Der Figure 2 37 Constrained Heat Equation imal Solution p tives at Opt Iva Second Partial Der Figure 2 38 70 2 7 Optimal Control Problems The main difference between optimal control and data fitting problems is the formulation of the objective function In the first case the objective function to be minimized is an arbitrary function depending on the solution of the underlying dynamical system often formulated as an integral whereas in the second case we minimize a sum of squares or any related norm as considered in the previous sections Another difference is that optimal control models contain so called control variables in addition to some discrete parameters that is a function s t JR to be modified until a suitable cost function is minimized However we have to approximate control variables so that they can be represented by a final set of parameters Typical control approximations are piecewise constant piecewise linear cubic spline or exponential spline functions either subject to a constant or a variable set of break points In the latter case the switching points are also optimization variables where the required serial order leads to a couple of additional linear inequality constraints A special form of control variable is called bang bang function i e functions defined by two co
376. reas and also of the boundary conditions for each individual equation The following options are allowed for the left and sepa rately for the right end point of the area 0 no boundary condition 1 Dirichlet type boundary condition i e function value given 2 Neumann type boundary condition i e derivative value given 3 both types of boundary conditions Discretization In case of addressing individual difference formulae to the state variables the following approximation schemes can be combined 1 3 point difference formula recursively for second derivatives 2 5 point difference formula recursively for second derivatives 3 forward differences for first derivatives 4 backward differences for first derivatives Spatial Positions of Coupled ODE s The positions of the spatial variable x where ordinary differential equations are coupled to the system of partial equations must be given The order must be increasing and any decimal value within the integration interval is allowed The number and order of entries must coincide with the number and order of ODEs defined in the model function file Decimal numbers are rounded to the nearest integer that describes a line of the discretized system Spatial Positions of Fitting Criteria The positions of the spatial variable r where fitting criteria and corresponding measurement values are set must be defined The order must be increasing and any decimal value within the integration i
377. related to Fortran but with emphasis on code flexibility and speed Basically there are two ways to implement automatic differentiation called forward and backward accumulation respectively Both are used in PCOMP one for the direct evalu ation of function and constraints values the other one for generation of Fortran code see Dobmann Liepelt and Schittkowski 115 for details The first variant was implemented for the parameter estimation codes we are interested in Note that a particular advantage of gradient calculations in reverse accumulation mode is the limitation of relative numerical effort by a constant that is independent of the dimension i e the number of variables A review of further literature and a more extensive discussion of symbolic and automatic differentiation is given in Griewank 186 An up to date summary of related papers is published in Griewank and Corliss 184 First we have to investigate the question how a nonlinear function is evaluated The idea is to break a given expression into elementary operations that can be evaluated either by internal compiler operations directly or by external function calls For a given function f the existence of a sequence f of elementary functions is assumed where each individual function f is real valued and defined on JR 1 nj m 1 fori n 1 m We define now the situation more formally by a pseudo program Definition 5 1 Let f be a real valued function define
378. reover we assume that weights w are given which can become zero in cases when the corre sponding measurement value is missing if artificial data are needed or if plots are to be generated for state variables for which E data do not exist The subsequent table contains the actual number I lt l of terms taken into account in the final least squares formulation The additional independent model variable c called concentration in the previous models is not taken into account for simplicity The system of partial differential equations under consideration is Uy Fi p U Ug en 0 2 t n Fn P U Uz Uze V Z t with state variable u ui Un We denote the solution of the system of partial differential equations by u p x t and v p t since it depends on the time value t the space value z and the actual parameter value p v denotes the additional coupled differential variable To simplify the notation flux functions are omitted Initial and boundary conditions may depend on the parameter vector to be estimated Since the starting time is assumed to be zero initial values have the form u p T 0 x uo p z and are defined for all z zr xg For both end points z and xp we allow Dirichlet or Neumann boundary conditions une uv p v t u p zg t uP p v t us p zr t t p u t Uz p tR t P p v t for 0 lt t T where T is the final integration time for example the last E time value t
379. right 0 C FUNCTION h h u C END C 15 Chapter 7 Problem Data and Solution Tolerances To start a parameter estimation or simulation run EASY FTT ModelDesign generates an input file for MODFIT or PDEFIT respectively The file has to contain all data that are passed to the numerical algorithm i e measurement values starting values for the parameters to be estimated solution tolerances etc We have to distinguish between general problem data that are independent from the type of the mathematical model and the model dependent information needed to start a parameter estimation run Data are collected within the main form of EASY FIT 4 sia and are kept in the data base until they are deleted or changed 7 1 Model Independent Information First a user has to provide database information that is independent from the underlying model Part of the data required is optional and only needed to document the input part is mandatory to set tolerances and options depending on the choice of an optimization routine 7 1 1 Problem Name The string to be inserted here is the unique key number to identify the parameter estimation problems in the database Thus the string must be non empty and unique It is not possible to change the name subsequently If one needs to alter the problem name one should copy the actual problem to another one with the desired name and delete the old one The problem name serves to define file names
380. riterion from 1 65 10 to 8 0 10 under the stopping tolerance 107 bounds as shown in Table 3 9 and the corresponding confidence levels are found in Table 3 8 They are significantly smaller than in case of the 40 measurements taken in the previous ny ti 0 183 0 0604 0 0073 0 03 0 074 0 0123 0 0068 na t na t 0 267 0 0022 0 0929 0 131 0 0934 ns t 0 0018 0 0111 0 171 0 036 0 0035 0 0025 0 0031 Nineteen weights are above the lower section and also the reduction of experimental expenses is significant 103 Chapter 4 Numerical Algorithms EASY FTT ModelDesign serves as a user interface for the parameter estimation programs MODFIT and PDEFIT that are also executable outside of EASY FIT 429 One of its features is the automatic generation of input files in ASCII format for the codes mentioned above Model functions are either defined symbolically to be executed by the automatic differentiation tool PCOMP or must be given in form of Fortran codes The corresponding data and code organization is documented in Chapter 8 In this chapter we describe very briefly the underlying numerical algorithms implemented 4 1 Data Fitting Algorithms The parameter estimation programs contain interfaces for a nonlinear least squares algorithm called NLPLSQ see Schittkowski 446 By transforming the original problem into a general nonlinear programming problem in a special way typical features of
381. rm p u yu we would like to identify Subsequently uniformly distributed random errors 5 are added to the data and we replace olu in 2 121 by a piecewise linear interpolation of pi pa see subsequent table u plu 0 0 0 0 2 pi 0 4 pa 0 6 pa 0 8 Pa We know that u does never exceed 0 8 in this case Constraints are defined to guarantee that the parameters remain monotone that pi Spo lt p3 lt pacl The least squares code DFNLP is executed where the underlying PDE is discretized at 21 lines Starting from the data given in the subsequent table DFNLP terminates after 6 iterations The obtained optimal parameter values are listed in Table 2 16 Obviously the constraints are satisfied and the first one becomes active The deviation of u from the exact coefficient function yu is shown in Figure 2 34 We conclude that an identification of p u is possible within the known experimental errors Figure 2 35 shows that the experimental data are fitted and Figure 2 36 plots the surface of the solution function u p z t It is also possible to define dynamical constraints where the restriction functions depend on the solution of the partial differential equation and its first and second spatial derivatives 66 Reactive Solute Transport 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 Figure 2 34 Identification Error Reactive Solute Transport 0 6 0 5
382. rol Counciol IEEE Service Center Piscataway NJ 2419 2422 Timmer J Rust H Horbelt W Voss H U 2000 Parametric nonparametric and para metric modelling of a chaotic circuit time series Physics Letters A Vol 274 123 134 Timoshenko S Goodier J N 1970 Theory of Elasticity McGraw Hill New York Tjoa L Biegler L 1991 Simultaneous solution and optimization strategies for parame ter estimation of differantial algebraic equation systems Industrial Engineering Chemistry Research Vol 30 376 385 Tjoa T B L T Biegler L T 1992 Reduced successive quadratic programming strategy for errors in variables estimation Computers and Chemical Engineering Vol 16 523 Troeltzsch F 1999 Some remarks on second order sufficient optimality conditions for non linear elliptic and parabolic control problems in Proceedings of the Workshop Stabilitat und Sensitivit t von Optimierungs und Steuerungsproblemen Burg Spreewald Germany 21 23 4 99 Tveito A Winther R 1998 Introduction to Partial Differential Equations Springer New York Tzafrini A R 2000 Mathematical modelling of diffusion mediated release from bulk degrad ing matrices Journal of Controlled Release Vol 63 69 79 Ulbrich S 1995 Stabile Randbedingungen und implizite entropiedissipative numerische Ver fahren f r Anfangs Randwertprobleme mehrdimensionaler nichtlinearer Systeme von Erhal tungsgleichungen mit Entropie Dissertation TU M
383. roximated by finitely many parameters Now we extend our model function bin t by the design parameters h p q t and assume that we have a set of experimental time values tg k 1 l Moreover we let Pin ai h p q t3 h p q 4 and denote by F p q V f p q the Jacobian matrix of f p q subject to p IR where q IR For simplicity we assume that F p q has full rank for all p and q A formal performance measure is available based on the covariance matrix C p q I p q 1 where I p q F p q F p q denotes an approximation of the Fisher information 90 matrix and where we omit a guess for the error variances of the measurements to simplify the notation In other words we assume that all experimental data are measured with constant error The volume of a confidence region for a given model parameter p IR is given by lp pp Ip a pp lt os 3 14 with a statistical parameter Q see 3 3 Formula 3 14 describes an ellipsoid and the goal is to minimize its volume on the one hand but on the other to prevent also degenerate situations where the maximum and minimum eigenvalue drift away This is to be achieved by adapting the design parameter q for a given model parameter p which is obtained either from a preliminary experiment literature or a reasonable guess Possible criteria are available either for C p q or p q respectively depending on the procedure how to measu
384. rs This is one of the main investigations when analyzing the output of data fitting Related problems are whether it is possible at all to identify parameters or how to eliminate redundant ones as will be discussed in the subsequent sections Another important question is experimental design where we want to create or improve existing experimental conditions The goals are to reduce the number of costly experiments to reduce error variances or to get identifiable parameters The corresponding tools are summarized in Chapter 3 A brief review of the numerical algorithms implemented is presented in Chapter 4 Only some basic features of the underlying ideas are presented More details are found in the references and in particular in Schittkowski 438 The codes allow the numerical identification of parameters in any of the six situations under investigation The executable files are called MODFIT EXE and PDEFIT EXE Nonlinear model functions can be evaluated symbolically Thus any compilation and link of Fortran subroutines is not required whenever model functions are defined or altered in this way A particular advantage of this approach is the automatic differentiation of model functions to avoid numerical truncation errors The corresponding program is called PCOMP see Dobmann Liepelt and Schittkowski 115 and is part of the executable codes The automatic differentiation algorithm the PCOMP language and error messages of the parser are descr
385. rsop ure SI eyep erguouodxo pourez3suo uorjn os RIOT swag orjyexpenb omg yya uro qo 1d serenbs 3seorq SULIO orrjourouoSLr YIM uro qo1d sorenbs jseor Surjjg eyep ergueuodxzq ums parenbs tan uro qoud sorenbs jseo T suLIoj re ur QZ uro qo1d sorenbs 3seo q SULIS orjerpenb yy uro qo1d sorenbs 3seo T SULIS orjerpenb YIM uro qo 1d sorenbs seo uorjoung 9599 erquouodx zq swo re ur mMoz yya ulo qoud sorenbs 3seo poure13suo uorjoung 9899 erguouodx zq SULIO OLIJOUIOUOSII pue eryuouodxo YIM uro qoad sorenbs spot swo u A s JIM uro qO Id sorenbs 4svo T uorjung s oMoq SULIO INOJ quara uro qo id sorenbs 3seo q 91q83sun AIS turo qo Id 3s93 oruopeoyv uorjoo1rp ex UL Ao eA estoy uro qo id sorenbs 3seorq suw 9914 q3r ulro qoid sorenbs spot uorjoung 9899 erquouodx zq uorjoung 9899 erquouodx q suorjoung erurouATOd oay uro qo1d sorenbs spot sw OM UM ulo qo1d sorenbs spot sw 9914 Yy1M uro qoud so renbs 3seo T puno b390q oo oo On Ch Ch CH Ch cn Ch CH op On On On cn On Ch OC Ch coo cco CC CH On OC CC Ch Ch CH Ch CH Oh CH o Ch Ch Ch Ch Ch CO CO Oh vi Gi Ch O O O O n OV qI TOG vr OT 0c 0c 0c TT I TT A CN CN CN CN cO cO lt bh lt lt SI NN CO cCoO CO CO Y B bh aO ao 0 CH VSEdL cG8Sd L IGEdL 08 amp d L FCL EdL GE dL A6 amp d L credL 808dL JOEL UCL 886d L 986d L GHG GLEd L 696d L 249cd L I9cd L 096d L 986d L SCd L AYcd L
386. rxtut pmbry sery ses jo pnop eorieqds e Jo 1orAeqoq ewy lo se3 jo Surypoes9 NJ LI soje d 007Z WLIS poo 3o ut jue3suoo ut 1oq1osqe ses oje d N soqeyd oz WLIS poof 3e ur jue3suoo TILA 1oqiosqe ses yerd N 1038 1980 e3seq ure r rurro 3 sgurids Aq po dnoo sosseur Jo soLIog sqjue d SULMOIS Jo surojs S 3001 ou ur mm jo peodg sqjue d SUIMOIS Jo surojs s 3001 94 ur mm Jo peo1dg SUIPUIQ ouere jo SOTJOUTY JJO UC 959 10 JO 141017 LL se onuoqri jo Surp ojo1 pue Surppoju LL eseopnuoqu jo Surp oJ91 pue Surp oju LL se onuoqri jo Surp ojo1 pue Surp oju LL se onuoqri jo Surp oJ91 pue Surp oju LL eseepmuoqu jo Surp ojo1 pue Surp oju uro qo1d uorjonpur oouooso1on 4 uro qo1d uorjonpur oouooso1on 4 sisoyyudsoyoyd Jo 938 1 oouooso1ong jseq dA 109 3uiooo posrourur YM pra 4 rreg axe Jo voryepndod ysta uy Sot e ur o1njedoduio puno4bDsovq oO c CO On ei OC CH Ch CC On On Ch CC Ch OC Ch Ch CC Ch Ch Ch CH 3 CO On ei CC Ch On Ch On On Ch CC Ch CH Ch cn CC occ Ch CH E NN O O Gi 00 0 o lt EIN 9 OC w ee x d ai cO a0 NN c n N co LG OF TIT GL 816 0 0 9 OV 001 001 00 IT IT VG OV 8 8 cv GL 69 GST 8E IT OT 0 lt ut SIN GO CQ r GO GO oO x aO aO p lt Go Go CN CN CN GO CN CY GO THSOONTO ASOONTO cODdOTO UYACTITO Irn4ggI5 cOIISVO TIOIISVO qn01O0SV5 TIO SVD saV SVO ISHV SVD IOSO d 4 TION
387. ryp 197em pUNOIS jo uorjeanj3eg soroods JO 324018 8919 Ape3e203n TYLA opour uorsngtp uorjoeo SISATODATS MOP 9 qisso1d uroour tum3uourour PUR Sseur JO uor eAJ9suoo YPM YYMOIS IILH uuin oo Y ur uorsngrp ses euorsuourrp ou uuin oo Y ur UOISNJIP ses p euorsuourrp ou UOI 29AUOO Ser uorsngtp TILA ro ur o qqnq ses snoostA uoN Im UeIssner e JO uorsngtp uorj29Auo JUO SurAour juejsuoo uou YILM opour uorjesedoad ourepq uorge ndod xoz Jo uornqunsrp somqeqq Ag SPA uorsngiq SUIT SurjreA YM erpour snozod qsnodq MOTT son eA VIPUL oye1ouoSop YM erpour snozod y3no1q7 MOTT 1039891 ILTNGNY Mop reurure Teurroq30sp Tepour uoryeg edoud oureg suopueg 194M T UOIjoeo1 RULIOYJOXS ouo YPM 103o o1 poq P XY onAqe3e7 1039891 poq poxY o1yATeyey uorjonpuoo 9AION puno4bDovq CO ooo CH CO CO CO CH oo oo oo On CC Ch Ch ch CC Ch CH o CO ooo CH ON N CH oo O CH lt H I oo oo CC Ch Ch Ch Ch CH o E gl pl ot e ed vd ri ri rz NNN Cl 4 74 c Cl MM Di ri r r r ri rd a OT 92 TT OT GE 9 Ic 0c c9 Gy ST S 2 O8T 09 681 0 0cI 08 8I EI 8y 9T Ise N CN iO ON 10 N cO CH g wo GO A Gi GO Go CN CN NN GO c c CH MO LVHH NOD LVHH AO LVAH dO LVAH dd LvuH gd LVH LV4AH HLMOYO MUNAOHO ODATO OHO V ISD CHIA SVO LI SVO ANOO SV gana svo TIH D INOW XOH INTA MWd MOTA aNd MOTA MOTA INV TA HHXIJ
388. s Initial Parameter Set Number of Parameters to Compute A Covariance Matrix and Confidence L ET a Intervals ast Compute Parameter Set Confidence Level 2 5 4 Ve Gradient Evaluation forward differences _y Order of Numerical Differentiation Tolerance 0 4 Figure 7 2 Simulation Tolerances typical features of a special purpose method are retained the combination of a Gauss Newton and a quasi Newton search direction in case of a least squares problem see Schittkowski 429 The additional variables and equality constraints are substituted in the quadratic programming subproblem so that calculation time is not increased significantly by this ap proach In case of minimizing a sum of absolute function values or the maximum of absolute function values the problem is transformed into a smooth nonlinear programming problem NLPLSQ 446 is capable to handle any additional linear or nonlinear equality or inequality constraints Since the number of variables increases with the number of experimental data least squares problems are treated as general nonlinear optimization problems if the number of data points exceeds 500 The total number of iterations might increase but the calculation time per iteration is decreased Min max problems are transformed into a general smooth optimization problem with only one additional variable and solved by the code NLPINF 448 A sophisticated active set strategy is applied in case of
389. s Lecture Notes in Mathematics No 1270 Springer Berlin Enright W H Hull T E 1976 Comparing numerical methods for the solution of stiff sys tems of ODEs arising in chemistry in Numerical Methods for Differential Systems L Lapidus W E Schiesser eds Academic Press New York 45 66 Farnia K 1976 Computer assisted experimental and analytical study of time temperature dependent thermal properties of the aluminium alloy 2024 T351 Ph D Thesis Dept of Me chanical Engineering Michigan State University Fathi H Schittkowski K Hamielec A E 2004 Dynamic modelling of living anionic solu tion polymerization of styrene butadiene divinyl benzene in a continuous stirred tank reactor train Polymer Plastics Technology and Engineering Vol 43 No 3 571 613 Fedkiw R P Merriman B Donat R Osher S 1996 The penultimate scheme for systems of conservation laws Finite difference ENO with marquina s flux splitting UCLA CAM Report No 96 18 Dept of Mathematics University of California at Los Angeles Feldman H A 1972 Mathematical theory of complex ligand binding systems at equilibrium Some methods for parameter fitting Analytical Biochemistry Vol 48 317 338 Fedorov V V 1972 Theory of Optimal Experiments Academic Press New York Fermi E Ulam S Pasta J 1974 Studies of nonlinear problems I in Nonlinear Wave Motion Lectures on Applied Mathematics AMS Vol 15 143 155 10 148 149 1
390. se from collagen matrices Preprint IAM Erlangen Nr 284 University of Erlangen D 91058 Erlangen Ramsin H Wedin P A 1977 A comparison of some algorithms for the nonlinear least squares problem Nordisk Tidstr Informationsbehandlung BIT Vol 17 72 90 26 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 Ratkowsky D A 1988 Nonlinear Regression Modeling Marcel Dekker New York Reich J G Zinke I 1974 Analysis of kinetic and binding measurements IV Redundancy of model parameters Studia Biophysics Vol 43 91 107 Rektorys K 1982 The Method of Discretization in Time and Partial Differential Equations Reidel Dordrecht Renardy M Rogers R C 1993 An Introduction to Partial Differential Equations Texts in Applied Mathematics Vol 13 Springer Berlin Rhee K I Sohn H Y 1989 The selective carbochlorination of iron from titaniferous mag netite ore in a fluidized bed Matallurgical Transactions B Vol 21B 341 347 Richter O Diekkrueger B Noertersheuser P 1996 Environmental Fate Modelling of Pes ticides VCH Weinheim Richter O Noertersheuser Pestemer W 1992 Non linear parameter estimation in pesti cide degradation The Science of the Total Environment Vol 123 124 435 450 Richter O S ndgerath D 1990 Parameter Estimation in Ecology VCH Weinheim Richter O Spickermann U Lenz F 1991 A new model
391. ser uw lasso wi hp ti c a Yi 2 k 1 see 2 3 Example 2 1 RAT_APP We want to fit some parameters pi pa so that the data of Table 2 1 are approximated by a rational function i pot h p t pi Q Pl E pst pa see Lindstr m 288 and Deufthard Apostolescu 112 There is no concentration parameter but we want to fit the outer measurement values exactly i e we define two additional non linear equality constraints gi p bin ti y and go p h p t y with l 11 There is only one measurement set and all weights are set to 1 Thus the least squares data fitting problem is min Xia Mp ti yi pe f i mini 0 92 p 0 When starting the code DFNLP of Schittkowski 429 from p 0 25 0 39 0 415 0 39 with a termination tolerance of 107 we get the solution vector p 0 1923 0 4040 0 2750 0 2068 after 10 iterations The final residual is 3 78 107 and the maximum constraint violation is 3 1 1079 The individual residuals and the relative errors are also listed in Table 2 1 Model function and data are plotted in Figure 2 1 Since the model function A p t c does not depend on the solution of an additional dynamical system we call it an explicit model function Otherwise h p t c may depend on the solution vector y p t c of an auxiliary problem for example an ordinary differential equation that is implicitly defined Models of this kind are consider
392. serve as starting parameters for the nonlinear programming algorithm applied Consequently we allow only index 1 systems unless it is guaranteed that consistent initial values for the discretized DAE are available Also any jumps or discontinuities at initial values of algebraic equations do not make sense 4 8 Statistical Analysis Proceeding from the assumption that the model is sufficiently linear in a neighborhood of an optimal solution vector and that all experimental data are Gaussian and independent some statistical data can be evaluated e Variance covariance matrix of the problem data e Correlation matrix of the problem data e Estimated variance of residuals e Confidence intervals for the individual parameters subject to the significance levels 1 5 or 10 respectively A priority analysis is performed after a data fitting run If a model seems to be overde termined the computed levels give an impression of the significance of parameters and help to decide upon questions like e which parameters can be identified e which parameters should be kept fixed e whether additional experimental data must be required Moreover overdetermined data fitting problems often lead to unstable and slow convergence of Gauss Newton type least squares algorithms To detect the significant parameters on the one hand and the redundant or dependent para meters on the other we apply the following procedure Successively parameters are eliminated
393. set Print flag indicates the desired information to be displayed on the screen 45 0 no output at all 1 only final summary of results 2 oneoutput line per iteration 3 detailed output for each iteration A value greater than 0 is only recommended in error cases to find out a possible reason for non successful termination Maximum Number of Iterations For computing consistent initial values NLPQLP requires the maximum number of iterations to be defined here A value of 50 is recommended Termination Accuracy To compute consistent initial values by NLPQLP one has to define the final accuracy by which the subproblem is solved In case of exact derivatives a value between 1 0E 8 and 1 0E 12 is recommended Consistency Parameters x A Output Flag for Subproblem i 5 i so Y Iteration Bound for Subproblem 50 y A Final Accuracy for Subproblem 1E 10 L Fe Figure 7 23 Consistency Constraints Restrictions are allowed for models based on partial differential equations and can be formulated in form of equality and inequality constraints with respect to the pa rameters to be optimized and the solution of the dynamical system at some of the given experimental time and spatial parameter values Where the total number of constraints can be retrieved from the subsequent table the number of equality constraints must be supplied Equality restrictions must be defined first in the input file for model functions T
394. side of a differential algebraic equation with respect to the system parameters since they have to apply Newtons method to solve certain systems of nonlinear equations The Jacobian is evaluated either numerically by the automatic differentiation features of PCOMP or must be provided by the user in form of Fortran statements In any case is possible that the Jacobian possesses a band structure depending on the DAE sys tem EASY FIT c Pes s allows to define the bandwidth that is passed to the numerical integration routines to solve systems of internal nonlinear equations more efficiently The bandwidth is the maximum number of non zero entries below and above a diagonal entry of the Jacobian and must be smaller than the number of differential equations When inserting a zero value it is assumed that there is no band structure at all Break Points Similar to ordinary differential equations it is possible to define additional constant break or switching points where the corresponding time values are to be defined in form of a separate table The integration is restarted with initial tolerances at these time values The numerical values inserted must vary between zero and the maximum experimental time value Time values are ordered internally If the table contains no entries at all it is assumed that there are no constant break points with respect to the time variable On the other hand it is possible that the last np optimization variables to be
395. soud Jo suor 5 9 o qTS19A9 uononpoad gq Hd 103 opour po1n3ona3g Tepour peordopooeurreud pequourjreduroo reour sojeurp rooo eorrouds ur uoryeor dde eorgnooeurreu q Uuorjoeor orureu p ooeurreu d osop o 8 uts suorjyeqosqe 3uoursos e YIM popou orjoup ooeurreud reour uoryei1jsrururpe sn oq Yy 9POUL orjeun ooeurreud reour I OI 9011 Se YHA 110139891 orureu p ooeurreq d TO OI eur Se UYLM uorgoeor orureuAp ooeurreu d I0 0 OI eurr Sep yya uorgoeoz orureuAp ooeurreu d OUIT BR erjytur o qerreA ouo YPM opour orureu p ooeutreuq sjuouiLrod X 8 10 ourrj ergrut poxy ouo YIM popou orureu Ap ooeutreuq puno4Dsovq CC CC Ch O O Ch Ch Ch Ch Ch coo cco Cc e CO CO Ch CO CU ec o CO CN O MAN CN O Hrd CGN O Le CC CC Ch CC Ch Ch Ch Ch Ch Ch Ch Ch cc Ch CO FF oo oo c e mo Go cQ cQ bf tu uL 8 89 I AT 9v 0 031 00 9 GOL T VG 8 GL 8G 09 GG OT O8T GG IT I9 I9 I9 GE GE GN i ENS ce pec OO IGN SO eo A CS o N dN Q ei i 0 O lt TNdOd UAINATOd NAXTIOdA cATOd TATOd INLATIOd OTH ONTd WONILV Td IINSV Id UN LNV Id NODOLOHd S OLOHd Ud OLOHd OLOHd q HdSOHd Val SOHd gSHd VINUVHd dV IWHVHd OVAA VHd GNIM VHd INDI VHd LNAG VHd 9NAG VHd SNAG VHd YNAG VHd ENAT VHd 9t Du 35 ponurjuoo IS gS IN a gS gS IS gS SS a SS uou gS a a a gS gS a a gS SS SS GS DIDP vez
396. squares norm ES 2 NICE 2 118 k 1 i 1 for the L norm and q max Op php ti wil 2 119 for the maximum norm where measurements t y are given i 1 k 1 7 As for ordinary differential equations additional nonlinear equality and inequality con straints of the form g p 0 j Em gt gj p 20 me 1 m are allowed These restrictions are useful to describe certain limitations on the choice of parameter values for example monotonicity Example 2 26 ISOTHRM2 We consider the identification of a nonlinear coefficient function in a transport process through porous media see Igler Totsche and Knabner 232 or Igler and Knabner 231 The advective dispersive transport equation is 2 120 Duzz qu p u 1 for 0 lt t lt 6 and 0 lt z lt 1 For the diffusion coefficient we choose D 0 1 and the Darcy flow velocity is set to q 1 u is the sorption isotherm we want to identify Initial mass concentration is zero u p 1 0 0 and Neumann boundary conditions are set nn Fu fO San E 0 u 2 121 with inflow concentration f t given by linear interpolation of the data 65 i D Di 1 0 5 0 565 2 0 7 0 565 3 0 8 0 751 4 0 9 0 879 Table 2 16 Starting and Computed Parameter Values t fH 0 0 1 0 1 0 0 5 2 0 0 1 3 0 0 0 10 0 0 0 Experimental measurement values are generated at 20 equidistant time values between 0 and 6 spatial value x 1 and a sorption isothe
397. st be required Overdetermined data fitting problems often lead to unstable and slow convergence of Gauss Newton type least squares algorithms The idea is to successively eliminate parameters until 3 9 is satisfied The cycle is terminated in one of the following situations 1 The smallest eigenvalue of the Fisher information matrix is smaller than the given confidence level 2 The parameter correlations are reduced by 25 3 None of the above termination reasons are met and all parameters have been elimi nated Level 1 corresponds to the first eliminated variables level 2 to the second etc The final level can be assigned to several parameters indicating a group of identifiable parameters EASY FIT Main Form Ele Edit Start Report Data Delete Make Utilities 2 Type a question for help mwy nur ModelDesign ASY FIT d pl Prof Klaus Schittkowski EA y B Actual Problem SOIL BARAR Gerani i e kl Version 5 00 Jan 2009 Model Data Experimental Data 0 1844122922 0 0 68093742 0 11 3424116 0 961156410 0 32 0652504 0 234737626 0 295201966 0 30 6792481 0 35 3734967 0 35117631 0 32 1486058 0 35417561 D 34 9604712 D 37 688465 D 41 804973 D 396411788 0 42 0553530 0 399448693 D 44 5985468 0 47 1950153 D 46 2611539 0 38 1070763 D 41 8804772 0 421645614 0 38 4971451 0 394873428 0 37 973105 D 39 9274512 0 40 064445 D 38 4813858 0 38 1497874 0 38 8899581 0 38 62397
398. st coincide exactly with the format needed for the input of data for executing MODFIT EXE or PDEFIT EXE respectively The model function file with extension FUN or FOR respectively is the same edited by the user from the database directly Alternatively it is possible to export problem data and to generate two text files in a directory specified by the user A file with extension DAT contains all data in precisely the same input format as required by the numerical programs MODFIT EXE and PDEFIT EXE to be able to execute these programs independently from EASY FIT cPess Another 1 EASY FIT Main Form File Edit Start Report Data Delete Make Utilities Type a question for help rx ModelDesign Prof Klaus Schiltkowski LtASY FIT 8 Actual Problem 50L a C 3 Version 5 00 Jan 2009 Model Data Experimental Data Model Model Type Information Diffusion of water through soil convection anddispersion SS Project Number Dem Unit for Values 1 nee Ecos system system of steady state equations heraa ET Unit for Y Values Names Laplace Laplace formulation of model functions with con centration values and internal backtransformation Date 14121398 Unit forZ Values fx Memo van Genuchten M T Wierenga P J 1976 Mass transfer studies in sorbing porous media 1 Analytical solutions Soil Sci Soc Am Journal Vol 44 892 898 explicit explicit model functions with concentr
399. step sizes SIAM Journal on Scientific Computing Vol 10 257 269 Hairer E Wanner G 1991 Solving Ordinary Differential Equations II Stiff and Differential Algebraic Problems Springer Series Computational Mathematics Vol 14 Springer Berlin Hald J Madsen K 1981 Combined LP and quasi Newton methods for minmax optimiza tion Mathematical Programming Vol 20 49 62 Hamdi S Gottlieb J J Hanson J S 2001 Numerical solutions of the equal width wave equation using an adaptive method of lines in Adaptive Methods of Lines A Vande Wouwer Ph Saucec Ph W Schiesser eds Chapman and Hall CRC Boca Raton Han S P 1976 Superlinearly convergent variable metric algorithms for general nonlinear programming problems Mathematical Programming Vol 11 263 282 Han S P 1977 A globally convergent method for nonlinear programming Journal of Opti mization Theory and Applications Vol 22 297 309 Hanson R J Frogh F T 1992 A quadratic tensor model algorithm for nonlinear least squares problems with linear constraints ACM Transactions on Mathematical Software Vol 18 No 2 115 133 Hao D N Reinhardt H J 1998 Gradient methods for inverse heat conduction problems in Inverse Problems in Engineering Vol 6 No 3 177 211 Harten A Engquist B Osher S Chakravarthy S R 1987 Uniformly high order accurate essentially non oscillatory schemes III Journal on Computational Physics Vol 71 231 303 Har
400. stic The techniques and part of the examples are also discussed in Schittkowski 441 A case study for a system of partial differential equations is found in Schittkowski 447 3 1 Confidence Intervals We proceed from a general nonlinear model in its simplest form n h p t e 3 1 see also 1 1 where we omit another possible dependency of the right hand side from the solution of a dynamical system without loss of generality h p t is our model function depending on a set of model parameters p IR and t IR is the independent model variable also called explanatory or regression variable The function h p t is supposed to be differentiable subject to p IR and at least continuous with respect to t It is assumed that there is a true parameter value p which is unknown and which is to be estimated by a least squares fit The response 7 JR is the dependent model variable The above formulation proceeds for simplicity from a scalar variable t Generalizations to multi dimensional regression variables are possible without loss of generality Also multi response models where y possesses arbitrary dimension can be considered see Seber and Wild 458 To estimate the true but unknown parameter value p from given experimental data t and y i 1 l we minimize the least squares function l s p Y h p ti y 3 2 i l over all p JR Let f denote the solution of this data fitting problem Then f
401. stimated are pi po pa and pa t is the independent model or time variable to be replaced by experimental data The fitting criterion is h p z t p4 z2 and we use the starting values 20 ps 2 pa and z9 t for solving the system of nonlinear equations The subsequent input file shows the parameters tolerances and measurement values used The data fitting code DF NLP needed 20 iterations to satisfy the stopping conditions subject to the tolerance given The corresponding data and function plot is found in Figure 9 4 C NEASYFITNproblemsNDYN EQ RECLIG10 2 Steady state system receptor ligand binding study Demo Schittkowski Simulation nMol Null log nMol NPAR NRES NEQU NODE NCONC NTIME NMEAS NPLOT 50 0 NOUT 0 METHOD 01 0 OPTP1 200 OPTP2 20 OPTP3 02 OPTE1 1 0E 09 OPTE2 1 0E 00 OPTE3 1 0E 02 ODEP1 ODEP2 ODEP3 ODEP4 ODEE1 ODEE2 1 II ki OQ O OQ O O E 3 0 0 0 OO O F o0 oo oo 23 ODEE3 pl p2 ps p4 SCALE 0 0 OE 1 OE 1 OE 2 OE 2 OE 3 OE 3 OE 4 OE 4 OE 5 OE 5 OE 6 NLPIP NLPMI NLPAC NDISCO pa F 0 F OI E OI kk OY Oe OI 0 0 0 0 1 0 10000 0 0 0 1 0 10000 0 0 0 100 0 10000 0 0 0 2 0 10000 0 II F 0 332 1 0 0 331 1 0 0 331 1 0 0 327 1 0 0 321 1 0 0 289 1 0 0 250 1 0 0 125 1 0 0 077 1 0 0 019 1 0 0 010 1 0 0 002 1 0 0 001 1 0 0 100 1 0E 11 0 The corresponding system functions must be pr
402. sume that the model functions hy p y 2 t c and g p are continuously differentiable functions of p k 1 r and j 1 m and that the state variables y p ti cj and z p ti cj are smooth solutions subject to p All test problems based on differential algebraic equations are listed in Table B 5 where constraint counts are omitted Al p numuoo 0S 0S 0S gS a gS IS a gS gS gS a ma Daag m 10 YN 030p L67 Let Let 34 sre pez lezz TF uorjerurroJ e xoput o durexo oruropeoy uorjeurloJ c xoput o durexo orupeoyv uomsemuroj T x pur o durexo orurpeoy Autre TIM FV Joddoo uo PIQIOSPe o1njsrour 10j opour 1oAe rpn A ouin oA juejsuoo q3r4 OULTJOW JO uorjesuopuo o1nj no uorsuodsns ut s 99 3ue d poj3e ost Jo uorjeArj n2 10jonpuooruiros 1s Te16 oO uurn oo uorje nsrp u593eq e 10 uorye nopeo urod qqnq puoq uoSo1p amp g uoSo1pKu ur uogoyd jo UOTZISURIT 1039891 oye USISOP JUOWILIOdXo suorjoeor 958 PUL MOTS 1039891 0310 eurioq30sq USISOP equourriodxo suorjoeor 958 PUL MOTS 1039891 0310 eurioq30sq SJOS GD 99 17 suorjoeor 988 PUL MOTS 1032 91 UIQ eurioq30sq SJOS VILVP OM suorjo or jsej PUL AO S 10J9891 U9I8Q eurioq30sq GT OPE 29s Jep ouo suorjoeor jsej pue MOTS 1032 91 VT RULIOYOST CT ETE 19S mp ouo suorjoeor jsej pue MOTS 1032 e91 qjeq RULIOYOST O Sap 00T 10 e1ep suorjoeor JSV pue MOTS 1039891 U9I8Q eurioq
403. tarting the integration If not consistent initial values must be computed by solving the above system of nonlinear equations subject to z where the initial values for the differential equations are inserted Example 2 6 BATCHREA We consider a simplified batch reactor model discussed by Caracotsios and Stewart 75 where 6 differential and 4 algebraic equations are given ji haj222 y 0 1 5776 Yo piy ye paz4 D31222 ye 0 8 32 Ya payaza payayo P523 y3 0 0 Ya paya4ye P523 s Yya 0 0 Ys piy2ye P224 ys 0 0 2 39 Ye P1Y2Y6 Paysye pa24 Ps23 eil 0 0131 0 z 0 0131 y6 22 23 24 2 0 2 0 pr 21 22 pm 2 0 2 0 Ds 1 23 Paus za 0 0 ps 21 24 Peys pg Sae 0 z 0 5 De Jp Apzyi 0 guarantees consistent initial values for the algebraic vari ables Measurement data are simulated for t i i 1 10 pj 1 1 8 with 3 correct digits and fitting criteria are yi ye The DAE is integrated by an implicit method with absolute and relative termination tolerance 10 and the least squares code DFNLP is executed with final accuracy of 1079 DFNLP terminates after 58 iterations with a scaled residual of 0 000029 Starting values p and final parameter values p are shown in Table 2 3 The data fitting model is highly overdetermined In other words we have too many parame
404. tems of the form Ut J p u u z t gt Us g p u v z t with initial values u p z 0 u p x v p 0 t p t We proceed from the general explicit formulation Oua e E F 9 U Ug Ugg Z L Bt alp 2 81 0 Falp u us Use Pe where x JR is the spatial variable with 1 lt z lt zr and 0 lt t T Initial and boundary conditions are the same as in the previous section see 2 72 and 2 73 But now the state variables are divided into ma so called differential variables ug u1 Ung and n algebraic variables ua Diaen Ungtna Where the number of algebraic variables is identical to the number of algebraic equations summarized by the vector Fa The dynamical system 2 81 is also written in the equivalent form Qui Fi p u ug ugg t t Bt ip Dis Ot Faal o Uros Z t E 2 82 0 E Frati P Uy gs yy 45 t gt 0 Fna na P U Uy Ura T t gt if we consider the individual coefficient functions F F Fan However we must treat initial and boundary conditions with more care We have to guarantee that at least existing boundary conditions satisfy the algebraic equations for example 0 Po p u p zr 1 u p zr t es PEE t ds t gt 2 83 0 Falp U p A A 0 A l where u is the combined vector of all differential and algebraic state variables If initial conditions for discretized algebraic equations are violated i e if equation 0 F p u p
405. ten A 1989 ENO schemes with subcell resolution Journal on Computational Physics Vol 83 148 184 Hartwanger C 1996 Optimierung von Antennenh rnern im Satellitenbau Diploma Thesis Dept of Mathematics University of Bayreuth Germany Hartwanger C Schittkowski K Wolf H 2000 Computer aided optimal design of horn radiators for satellite communication Engineering Optimization Vol 33 221 244 14 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 Haug E J 1989 Computer aided Kinematics and Dynamics of Mechanical Systems Allyn and Bacon Hayashi H 1989 Drying technologies of foods their history and future Drying Technology Vol 7 315 369 Hayes B T Lefloch P G 1998 Nonclassical shocks and kinetic relations finite difference schemes SIAM Journal on Numerical Analysis Vol 35 No 6 2169 2194 Hearn A C 1978 Reduce user s manual Version 3 3 Rand Publication CP78 Santa Mon ica USA Hedrich C 1996 Modellierung Simulation und Parametersch tzung von K hlprozessen in Walzstrafen Diploma Thesis Dept of Mathematics University of Bayreuth Germany Heidebrecht P Sundmacher K 2003 Molten carbonate fuel cell MCFC with internal reforming Model based analysis of cell dynamics Chemical Engineering Sciences Vol 58 1029 1036 Heim A 1998 Modellierung Simulation und optimale Bahnplanung von Industriero
406. ters making it impossible to eval uate them at least from the given data Only some of them are in the order of the known exact values 16 0 i p D 1 3 0 1 031 2 1 5 0 751 3 1 1 0 591 4 0 5 0 946 5 0 6 1 029 6 1 4 2 983 7 10 0 9 752 8 0 1 1 068 Table 2 3 Starting and Computed Values for Example 2 6 2 5 3 Switching Points There are many practical situations where model equations change during the integration over the time variable and where corresponding initial values at the switching points must be adopted A typical example is a pharmacokinetic application with an initial infusion and subsequent application of drug doses by injection It is even possible in these cases that the solution becomes non continuous at a switching respectively break point We assume for simplicity that the dynamical model is given in form of an ordinary differential equation with initial conditions In a similar way we may define switching points for steady state systems or differential algebraic equations We describe the model by the equations Y Fpyto 000 pc Bas 2 40 in PRswyto r T E for 0 lt t lt r and Figg to cd nien wm Sg 2 41 Um Fan p Ee H um u po n 759 for T lt t Tj44 1 np ny is the number of break respectively switching points 7 with 0 lt 71 lt lt Tp lt T where 74 44 T is the last experimental time The initial values of each subsystem are given by functi
407. th a residual of 0 22 107 see Figure 9 6 which was evaluated for a finer grid with 25 discretization points More examples how to formulate model functions and boundary conditions are found in Dobmann and Schittkowski 117 44 data 0 7 0 6 0 5 0 4 0 3 0 2 0 1 01 0 15 0 2 025 03 035 04 045 0 5 0 05 Figure 9 5 Final Trajectory for Heat Conduction Problem model function Figure 9 6 Surface Plot for Heat Conduction Problem 45 Chapter 10 Test Examples The reason for attaching a comprehensive collection of test problems is to offer the possi bility to try out different discretization procedures differential equation solvers and data fitting algorithms The problems can be used for selecting a reference problem when trying to implement own dynamical models or to test the accuracy and efficiency of numerical algorithms for example for comparisons with other methods In many cases parameter estimation problems are found in the literature or are based on cooperation with people from other academic or industrial institutions In many other cases however differential equations are taken from research articles about numerical simulation algorithms and are adapted to construct a suitable data fitting test problem Thus some model equations do not coincide exactly with those given in the corresponding references and the numerical solution is somet
408. the following table Note that the corresponding text is displayed if the error routine SYMERR is called with parameters LNUM and IERR In the version implemented for the parameter estimation codes an error is reported when starting the execution of a numerical algorithm i e when the parser analyzes the code The corresponding error code and a line number are displayed and a user should edit the PCOMP code before trying it again 1 file not found 2 file too long 3 identifier expected 4 multiple definition of identifier 5 comma expected 6 left bracket expected 7 identifier not declared 8 data types do not fit together 9 division by zero 10 constant expected 11 operator expected 12 unexpected end of file 13 range operator expected 14 right bracket expected 15 THEN expected 16 ELSE expected 17 ENDIF expected 18 THEN without corresponding IF 19 ELSE without corresponding IF 20 ENDIF without corresponding IF 21 assignment operator expected 22 wrong format for integer number 23 wrong format for real number 24 formula too complicated 25 error in arithmetic expression 15 26 27 28 29 30 31 32 33 34 35 36 45 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 internal compiler error identifier not valid unknown type identifier wrong input sign stack overf
409. the reaction of urethane The model consists of three differential and three algebraic equa tions and becomes more complex because of additional nonlinear equality and inequality constraints There remains the question how the techniques described in Section 3 3 can also be used for locating experimental time values Especially in case of time expensive experiments it is highly desirable to minimize their number and to conduct experiments only within relevant time intervals Thus we apply the same strategy outlined before but add artificial weight factors to the observations at a predefined relatively dense grid specified by the user These weights are considered then as design parameters A particular advantage is that derivatives subject to weights are obtained without additional computational efforts Section 3 4 contains the corresponding analysis and again the urethan example now with thew aim to reduce the number of experiments The techniques described so far do not depend on any special structure of the mathe matical model The only assumption is smoothness of objective function and constraints i e these functions should be twice continuously differentiable subject to the model and design parameters All examples with practical background consist of ordinary differential 80 or differential algebraic equations since the imbedded solution process generates additional numerical noise making numerical results and conclusions more reali
410. thout any safeguards we get an unstable solution as shown in Figure 2 9 only for x1 t Numerical instabilities occur also for x t Zaff and x t However when reducing the influence of the friction coefficient to u 0 01 we are able to integrate the system despite of the discontinuity see Figures 2 10 and 2 11 The dotted lines represent x t and x t respectively We generate artificial exact measurements for x t xa t 2 t and x2 t at time values t d fori 1 10 and p u 0 01 without any random errors Although the initial residual is in the order of 10 we are nevertheless able to recompute this optimal solution when starting from po 1 5 see Table 2 4 case 1 The least squares code DF NLP is applied with termination tolerance 107 From the differences of the velocities in Figure 2 11 it is obvious that we do not have more than two switching times Thus we add two additional optimization variables Tj and Ta leading to the differential equations miii f ulti t xi 0 1 1 fot ulii 22 Sall i 2 47 247 mail for all t lt 7 mo fit wld AO elen HO HOT uus mot fo u d a3 23 0 ah p ri 23 0 iip 71 for allt lt t lt r gt and moi forn B 23 0 s in Ta 4380 Bp for all t gt 72 Now the numerical instability for large friction coefficients mentioned above is avoided see Figures 2 12 and 2 13 where the estimates T 6 and Ta
411. time interval 0 lt t lt 71 In a similar way we adapt the Dirichlet boundary values u p 0 0 stemm u p0 2 0 0 lt t lt m p 0 t p 0 t 2 114 ul p 0 t zo nz E u p 0 t za mE gt see 2 106 Transition functions between both areas are the same as before i e ul p 0 5 t bu p 0 5 t l l 2 115 u2 p 0 5 t wt bd see also 2 115 and 2 108 Experimental data are generated in same way as for Exam ple 2 23 subject to parameter values of Table 2 13 As before each integration area of the partial differential equation is discretized by 21 lines a fifth order difference formula is applied the discretized system of 99 ODE s is solved by an implicit method with a restart at t T and the least squares algorithm DFNLP is started with the same solution tolerances After 109 iterations the optimal solution listed in Table 2 15 is obtained see Figure 2 33 for the final surface plot Obviously we are able to 63 p Po p D 1 0 5 0 1 09287 D 1 0 0 2 0 97516 a 1 0 10 0 1 02542 b 2 0 1 0 1 97548 Q 0 2 1 0 0 19998 Table 2 15 Exact Start and Computed Solution Heat Equation ulp tt p gt ZT Lor L de 2 7 L EEE D 1 5 lt LEAL 1 p i ELLER d y N i V M o al M YW A Q NX Ny ut WN AW NY AN M i Q WA d h NI AW AN WY AU y d
412. tine and fully documented by initial comments Since most applications will probably execute the corresponding main program we describe only the format of the input data and the usage of the subroutine required for model evaluation More technical information can be retrieved from the initial comments of the main program for example about linking parameters common blocks etc An input file named MODFIT DAT must contain the parameter estimation data some problem information and optimization data in formatted form The first 6 columns may contain an arbitrary string to identify the corresponding input row if allowed by the format 1 a80 FILE Name of output files generated by MODFIT The string may begin with a path name but must not contain an extension Suitable extensions are selected by MOD FIT 2 a6 4x 15 MODEL Problem name passed to subroutine SYSFUN for iden tifying data fitting models The subsequent integer de fines the general model structure 1 explicit model function 2 Laplace formulation of model function 3 steady state system of equations 4 system of ordinary differential equations 5 system of differential algebraic equations ud INFO Long Long information string for plot output string for Long information string for plot output output PROJECT Plot output first line of information block e g project number Plot output second line of information block e g mea surement characterization
413. tional vari ables NLPQLP computes a solution in seven iterations with termination accuracy 107 The total number of experiments is reduced to 5 see Figure 3 10 and would have to be taken into account only for Cs Model parameter Kp 10 is estimated subject to a confidence level 0 16 and K 0 1 subject to 0 00011 Input feed is very similar to the optimal feed of Example 3 4 The location of the time values seem to be exactly at the critical points which determine the structure of the dynamical system Example 3 7 URETHAN1 2 We consider again the urethane problem of Example 3 5 because of its practical relevance see also 3 21 It is pointed out in Bauer et al 29 that the experiments are expensive and that it is highly desirable to reduce their number as much as possible We proceed from 40 equidistant time values between 0 and 80 for the four 102 Table 3 9 Optimal Weights for Urethane Problem ti 12 16 18 20 26 28 30 32 34 36 38 40 80 CONDO KR WY s ki ki ka Q bi k C 3 measurable output functions ni n3 n4 and ns and try to reduce their number to only the significant ones without loosing the desired identification option as computed in the previous section It is to be noted that all substrates are measured independently of each other i e we have a total of 160 experimental data from where relevant ones are to be extracted The optimization routine needs 132 iterations to reduce the performance c
414. tions of p from the true m value lead to an exponential increase of the computed solution because of the hyperbolic functions involved Other reasons for considering the multiple shooting approach are singularities prevent ing an integration over the whole interval or highly oscillating solutions where the initial trajectories for starting the least squares algorithm are far away from the experimental data 30 Notorious Example Figure 2 16 Single Shooting Multiple shooting was first developed for solving boundary value and optimal control problems see Bulirsch 63 Bock 51 or Deuflhard 111 A brief outline is also found in As cher and Petzold 11 see also Ascher Mattheij Russel 10 or Mattheij and Molenaar 324 Multiple shooting is applied to the solution of data fitting problems by Bock 52 The basic idea is to introduce n additional shooting points along the time axis say Ua E LII Tu 2 58 where T t is the last experimental time value For formal reasons we define 7 0 Integration is performed only from one shooting point 7 to the next one 7 and then initialized with a shooting variable s IR i 1 ns m denotes the number of differential equations The differences of shooting variables and solution at right end of the previous shooting interval lead to additional nonlinear equality constraints In a more formal way we pr
415. to insert a suitable concentration value 3 Time and concentration dependent models To overcome the drawback mentioned above we assume that the dynamical model say an ordinary differential equation depends on an additional in the statistical sense independent parameter c i e c may be inserted into initial values right hand sides fitting criterion or even constraints Now we proceed from measurements of the form Met EE Linde 2 10 The model function is again given in the form A p t c and we want to estimate the parameter vector p by minimizing mint Diy Alp ti cj yi pel 2 11 Now we get the fitting criterion fato h p Esa cj Yij gt t 1 TES ly 3 1 p le 2 12 In the subsequent sections we proceed from the most general situation 2 11 and illus trate our approaches by examples 2 2 Explicit Model Functions In this section we restrict our investigations to parameter estimation problems where one vector valued model function is available in explicit form the so called fitting criterion with one additional variable called time and optionally with another one called concentration We proceed now from r measurement sets given in the form ye EE lig E RES M UN 2 13 where l time values le concentration values and l ller corresponding measurement values are defined Together with a vector valued model function h p t c hi p t c h p t ett we get a data fitting
416. trated by an example Consider the function f p p for n 1 A syntactically correct formulation would be IF p EQ 1 THEN f 1 ELSE f p 2 ENDIF In this case PCOMP would try to differentiate both branches of the conditional statement If p is equal to 1 the derivative value of f is 0 otherwise it is 2p Obviously we get a wrong answer for p 1 This is a basic drawback for all automatic differentiation algorithms of the type under consideration PCOMP allows the execution of external statements that must be linked to PCOMP in a special way see Dobmann Liepelt Schittkowski and Trassl 116 A frequently needed computational value in case of a PDE model is the integral with respect to the spatial variable z i e a J ui p z t dz j 1 where the integral is taken over the j th area where the PDE is defined 7 1 ny Index i denotes the th solution component we want to integrate i 1 ny The integral is evaluated by Simpson s rule and denoted by 13 SIMPSN I J in the PCOMP language This name can be inserted in an arithmetic expression for example to compute a fitting criterion The corresponding time value is either a measurement value or an intermediate value needed for generating plot data 14 5 3 Error Messages of PCOMP PCOMP reports error messages in the form of integer values of the variable IERR and whenever possible also line numbers LNUM The meaning of the messages is listed in
417. tration If more than one time and more than one concentration value is given or if partial differential equations are integrated then a three dimensional surface plot is generated where the time variable corresponds to the first horizontal x axis the concentration or spatial variable to the second horizontal z axis and the function values to the vertical y axis Surface plots show either the fitting function in case of concentrations or the model function i e the solution function of the partial differential equation in the other case In case of more than one measurement set plots are repeated for each set and are displayed in the order in which the data sets are defined Overlays are admitted for the internal graphics system and for GNUPLOT where the number of overlays is determined by the user Residual plots are introduced to detect visually systematic deviations of experimental data from the model function The plots show the individual deviations in form of two dimensional graphs The horizontal axis displays the corresponding serial number not the actual time or concentration value respectively In case of several measurement sets the plots are repeated for each set and are displayed in the order in which the data sets are defined The original output of the selected least squares algorithm is directed to a file depending on the chosen print level Subsequently the output can be displayed on request However one has to be a bit fami
418. tween the input file and the data fitting code For defining variables we need the following rules 1 The first variable names are identifiers for the n independent parameters to be esti mated i e for p Pn 2 If a so called concentration variable c exists then a corresponding variable name must be added next 3 The last variable name identifies the independent time variable t for which measure ments are available 4 Any other variables are not allowed to be declared Similarly we have rules for the sequence by which model functions are defined 1 First r fitting criteria hi p t c h p t c must be defined depending on p t and optionally on c 2 The subsequent m functions are the constraints gi p 9m p if they exist at all They may depend only on the parameter vector p to be estimated 3 Any other functions are not allowed to be declared 1 The constants n r and m are defined in the database of EASY FIT 4 In addition to variables and functions a user may insert further real or integer constants in the function input file according to the syntax rules of PCOMP Example To illustrate the usage of symbolic function input we consider an example We have two explicit model functions hi p t Dexp ht kD ho p t 5 exp kat exp hit The corresponding input file is the following one G l
419. uent table the number of equality constraints must be supplied Equality restrictions must be defined first 23 CO Number of Break Points to be PT A Optimized p 4 Constant Break Points Figure 7 12 Break Points in the input file for model functions The table allows to define time and concentration values for which solution values of the underlying dynamical system can be inserted 24 Order Serial order number of constraints Equality constraints must be defined first Name Arbitrary name for the constraint to be printed in reports Time Corresponding time value at which a constraint is to be evaluated Note that the time values are rounded to the nearest experimental time value to avoid a rest of the integration In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does not exist c x value Corresponding concentration value at which a constraint is to be evaluated Note that concentration values must be rounded to the nearest experimental concentration value In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental concentration value does not exist Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or concentration parameter The number of lines in the table m
420. ues L norm In this case the sum of absolute values of all differences between model function and measurement values is minimized L4 norm The maximum of absolute values of all differences between model function and measurement values is minimized 12 7 2 Model Dependent Information For each of the parameter estimation models of EASY FIT ode Design additional data are required depending on the model structure as outlined in Chapter 5 7 2 1 Model Data for Explicit Functions For the execution of the numerical analysis program MODFIT with explicitly given model functions we need some integers that cannot be retrieved from the model function file or other data Number of Measurement Sets The number of measurement sets must coincide with the num ber of data sets as given in the input table for experimental data Note that a data set corresponds to two input columns in the table for values and weights Number of Concentration Values The number of concentrations must coincide with the number of concentrations as given in the input table for experimental data If the value inserted is positive and the PCOMP input language is used then a concentration variable must be declared in the model function file If 1 is inserted it is supposed that the fitting criteria depend on an additional concentration variable and that one concentration value is assigned to each time value Explicit Model Parameters x A Number of M
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422. unction at intermediate spatial grid points The choice of the corresponding stencil depends on the magnitude of divided differences to direct the stencil away from discontinuities To solve also systems of hyperbolic equations a full eigenvalue eigenvector decomposition of the Jacobian of the flux function is performed with respect to u and the scalar ENO method is applied to coefficient functions after a suitable transformation Flux splitting at the cell walls is applied requiring separate decompositions for the left and right approximation see Donat and Marquina 120 and Marquina and Donat 321 The wind direction is estimated by the corresponding eigen value The resulting system of ordinary differential equations can be solved either by an implicit or explicit ODE solver as before or by a special Runge Kutta method with fixed stepsizes to satisfy the CLF condition Whenever a boundary condition in Dirichlet form ul p zr t urn t ue punt up 4 5 ui p ep ch p a xp t t l uk p Tio t m Cra aD um Tio t t is given for 1 lt k lt np then we know the value of the boundary function and use it to interpolate or approximate the function u p x t as described above In other words the corresponding function value in the right hand side of the discretized system is replaced by the value given Alternatively a boundary condition may appear in Neumann form uj pmnt g p t uka p za t tg p t 4 6 u
423. unctions Again we proceed from a system of ng differential and n algebraic equations in explicit formulation 2 81 where the state variables consist of nq differential variables uy and na ug Ua But now we introduce an additional flux function f p u Ue x t i e we suppose that our dynamical system is given by algebraic variables ua i e u 2 87 Falp J p u Ur T t fa p u Us T t u Ux Uy T t Oud Ot Falo f p u Ug X t f Dp u Ug X t u Ug Ure T t H n and 0 lt t T Initial and boundary lt m up where x JR is the spatial variable with x d conditions are the same as before see 2 72 and 2 73 Flux functions are useful in two situations First they facilitate the declaration of highly complex model functions given by their flux formulations In these cases it is often difficult in analytical form and one has to apply a first order discretization scheme to the entire flux function or impossible to get the spatial derivatives Example 2 19 MOL_DIFF The model describes the diffusion of molecules where a flux function is given by Dexp B c 1 es J p C Cr dt and the diffusion equation is Dexp 8 c 1 cz 9 Ox 48 a Zeche C Cy L t k Tk Ui 1 218 0 8865 2 182 0 8737 3 145 0 8609 4 109 0 8511 5 72 0 8063 6 36 0 6891 T 0 0 6678 8 36 0 6227 9 72 0 6366 10 109 0 6530 Table 2 10
424. uondiospe Xpn3s LSIN uondaospe Xpn3s LSIN uondaospe U1933ed SUXIJA U1993ed SUXIJA U1933ed SUXIJA U1933ed SUXIJA Iemm owo ouo Tepnoo ourouo q Tepnoo ourouo q Tepnoo ourouo A SOUT uormn oAo som um uoquo A st oeqor o Xpn3s LSIN uorssox8o1 reouruou erjuouodx q Xpn3s LSIN uorssox8o1 reouruou erjuouodx q amp pn3s LSIN uorsso1go1 reoutr uou Tom eyeP 091 3exreur orurouooo orureuA q jox reur orurouooo orureuA q uorje noir ur sojou ouoino JO SOLIOS SUIT 2TUIOUO290O9 AI s sop oorgj pue uoryeordde snjoq On popou 3uourjreduroo 1eourT osop o gurs pue uorjeor dde snjoq tw opour quourjreduroo reourTq ros ur UOLMIOS ureurop oundo Ul Me eriojeur 2rjse ooOstA 1e9urT Son eA OUT UL SIOLIO YPM SUL vjep reourT our 38 pue sjuourjreduroo e YIM opour orjourqooeurreud reour uorpnpuoo jeoq 1eourT rejnoseA 1 xo uorjedjsrururpe SSOPIAOIN 1eynoseA 1 X9 uoryexjsrururpe osoprapur yya squourjreduroo reourT osop o gurs uorgexjsrururpe sn oq Ya sjuourjreduroo TROUT puno4bDovq NM CH CC Oh Oh ei CH On On On CC Oh CH CC Ch O Ch Ch coco coc Ch CH mm cO OOOO ei CH c cc OOOO CC Ch OOO ei Ch CC Ch CH OT 8 LG Eg 8E VI VI VI VI GLI GI EE 9T IT OGL 001 981 84 8 ve v8 0 GE SOT 6I F OT DA A OE AE SG A A Er ES N HN e e o c c co a ALIV LdOIN GONOIN FIN XIN LVd XIN GLVd XIN LLVd XIN CT VSSTIIN OTVUSIN gdIVdSIN VIVUSIN ANIN L
425. upper bound for estimated parameter If the file RES contains the results of a previous run then the corresponding parameter values are read in and replace the given ones i e the PAR values Hacc input of NTIME lines for i NTIM E with the following data ti i th measurement time not smaller than zero g wE measured data i e experimental output and inidividual weight factor Gs measurement number k k 1 NMEAS a6 4x 15 INTEG Flag for evaluation of integrals with respect to spatial variable X 1 or not 0 ws a6 4x 15 SCALE Scale for weight factors 0 no additional scaling of weights 1 divide each weight factor by square root of sum of squared measurement values 1 divide each weight factor by absolute measurement value 2 divide each weight factor by squared measurement value 36 a6 4x 15 ORDER Order of partial differential equation variable X ORDER 1 Only first derivatives hyperbolic ORDER 2 First and second derivatives parabolic a6 4x 15 NLPIP Print flag for inner nonlinear equation solver NLPQLP executed to get consistent initial values in case of addi tional algebraic equations NLPIP 0 no output NLPIP 1 only final output NLPIP 2 one line per iteration NLPIP 3 extended output every iteration NLPMI Maximum number of iterations for NLPQLP e g 50 a6 4x g10 4 NLPAC Convergence tolerance for NLPQLP e g 1 d 12 a6 4x 15 NBPC Number of
426. ur jo uornjos yor dxa St ggep eryuouodxzq punosbyanq D Ch c Ch Ch CH CO On On On On On On On Q Q nn Ch Ch Ch Ch CC CH o CO Ch oc o0 c CH CO On On CC On On On CC Q Q Ch Ch Ch Ch CC CH o 086 OSG OSG 966 6c TOF LG SOP 0 0 00 08 GO GO OO o m o AA NA CN Po OTH 0 ON A Gi N O N gr oco goud OY UVA ESSAVO cSSNVD TSSAVO UV SSAVO qe ssnvo SSnvo SVININVO HGOAHMHG 4 WYAL dxXa LSL dxa LSHL dXW IdNS dx Ad dXM 9d dX Sd dXM Yd dxd Cd dXM Id dXM ALIT dX4 9 LIA dX SLIT dX4 PLIJ dX4 ELIT dX GLIT dX IWDU ponurjuoo A uorsso1do1 IBIUIUON 0 0 9T z ATI a 897 pnis LSIN uorssoidor reouruou peuouodxy Q 0 vo 9 ESOZONV I a 897 pnis LSIN uorssoidor reouruou peuouodxy Q 0 vo 9 CSOZONV I a 897 Apnys LSIN uorssoigor reouruou peuouodxy Q 0 vo 9 ISOZONV I X HOI Iosnequro S IOIOUI OSIQIM 0 cor ONAT SM d Apnys LSIN 9dooso1orur uorjoo o Suruueos Q 0 ISL GAN a orl ss u yu drp 0 u urd eud e jo uoryezueust ema 0 0 OF X UANOSI a eesi lese uoneuHoppoqre uo 0 0 41 HOO NOWUI d uoneurp qooq 1eo uor D 0 8 6 TOO NOUI IN pos Tepour ergrouodxo oorqj yndyno ynduy 0 0 TOG AXHEOI 0S uorenurioj q3oouirs uou snonurjuoo uou ose sourjnoi uorjye od oju I 0T e IOd LNI SS 928 sorureudp uoremdoq 0 0 X DHLNI 0S suonn os Ae Atom 0 Ic ALINIANI x siojyourered urew tuorunj 3899 pouonrpuoo T
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429. ust coincide with the number of constraint functions defined on the model function input file either Fortran or PCOMP 25 Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 13 Constraints 26 7 2 5 Model Data for Differential Algebraic Equations The code MODFIT solves parameter estimation problems based on differential algebraic equations see Hairer and Wanner 199 for details For the execution of MODFIT we need some data that cannot be retrieved from the model function file or other sources Number of Differential Equations Define total number of differential and algebraic equa tions The number of DAE s must coincide with the number of model functions for the right hand side and the number of initial conditions on the Fortran or PCOMP input file Number of Measurement Sets The number of measurement sets must coincide with the num ber of data sets as given in the input table for experimental data Number of Concentration Values The number of concentrations must coincide with the number of concentrations as given in the input table for experimental data If the value inserted is positive and the PCOMP input language is used then a concentration variable must be declared in the model function file If 1 is inserted it is supposed that the fitting criteria depend on an additional concentration vari
430. ution of the dynamical equation at the corresponding fitting point and its derivatives where k denotes the index of a measurement set we use the notation hy p t help u p zi t us p xy t uso p k t t 2 74 Each set of experimental data is assigned a spatial variable value x rr xg k 1 r where r denotes the total number of measurement sets Some or all of the z values may coincide if different measurement sets are available at the same local position Since partial differential equations are discretized by the method of lines the fitting points x are rounded to the nearest line Also in this case we assume that the observation times are strictly increasing and get the objective functions 35 Y wh help ti m 2 75 k 1 i 1 for the least squares norm x Y wi help ti vf 2 76 k 1 1 1 for the L norm and k Jik nr max p Wi elp ti 1 2 77 for the maximum norm Example 2 17 HEAT_A To illustrate the standard formulation we consider a very sim ple parabolic PDE the heat equation Ue ras 2 78 with a diffusion coefficient p gt 0 The spatial variable x varies from 0 to 1 and the time variable is non negative i e t gt 0 The initial heat distribution for t 0 is u p x 0 po sin zz 2 79 for all x 0 1 and Dirichlet boundary values u p 0 t u p 1 1 0 2 80 for allt 2 0 are set It is easy to verify by insertion that u p z t po ex
431. ver possible for example for explicit Laplace and steady state models calculation of consistent initial values or for dif ferentiation of right hand side of an ODE DAE subject to state variables If a model is implemented in Fortran it is recommended to provide all gradients analytically i e to generate them by hand However derivatives are not available analytically for the outer parameter estimation algo rithms in case of differential equations The only exception is the usage of internal numerical differentiation when executing the explicit solver In all other cases derivatives are approx imated by numerical differences The following derivative calculations are available 0 analytical differentiation 1 forward differences 2 two sided differences 3 five point approximation formula It is important to know that forward differences require n two sided differences 2n and the five point formula 4n additional integrations where n is the number of parameters to be estimated If PCOMP is used the value 0 should be inserted in case of explicit Laplace and steady state systems in all other situations a positive value Numerical Differentiation Error The corresponding tolerance needed to compute these ap proximation is defined separately in form of an integer that estimates the number of correct digits To give an example an entry of 5 leads to a perturbation of 10 for approximating the gradients If this parameter is s
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433. which constraints are to be defined Fe Order Serial order number of constraints Equality constraints must be defined first Name Arbitrary name for the constraint to be printed in reports Time Corresponding time value at which a constraint is to be evaluated Note that the time values are rounded to the nearest experimental time value to avoid a complete re integration of the system In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental time value does not exist c x value Corresponding concentration value at which a constraint is to be evaluated Note that concentration values are rounded to the near est experimental concentration value In case of doubt insert dummy experimental data with zero weights if constraints are to be defined at points for which an experimental concentration value does not exist Note that equality restrictions must be defined first and that dummy values must be inserted for each constraint not depending on the time or concentration parameter The number of lines in the table must coincide with the number of constraint functions defined on the model function input file either Fortran or PCOMP 18 Constraints xj T A Number of Equality Constraints ii A pie c x value 0 0 0 0 0 Figure 7 9 Constraints 19 7 2 3 Model Data for Lap
434. which the variables are defined must coincide with the order in which they are used in the model function part i e either a Fortran code or a PCOMP input file To guarantee the correct order in the underlying database we need corresponding order information Thus the first column must contain the sequel number of the variables After a successful data fitting run the optimal parameter values and if a positive sig nificance level has been set also their significance levels are listed Since all numerical executions are run as separate processes their termination is checked by the user interface only at special occasions e g in case of generating a report To see these data immediately after execution of MODFIT EXE or PDEFIT EXE one has to click on the command bottom update table 7 1 4 Input Type of Model Functions Basically there are two possibility to define the nonlinear model functions PCOMP All variables and functions are declared in a Fortran similar syntax where the order of the variables must coincide with the order used in the database For function input one has to follow the guidelines outlined in Chapter 5 Particular advantage is the possi bility to let derivatives be computed automatically during run time i e without additional approximation or round off errors Fortran Alternatively a user has the choice to prepare model functions in form of For tran statements if the PCOMP syntax is too restrictive or if
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436. xperimental Design It is outside the scope of this documentation to present a review of statistical methods that are available today to analyze data and results There exists a broad area in statistics called nonlinear regression or parameter estimation where these techniques are developed in detail see for example the books of Bard 27 Beck and Arnold 32 Draper and Smith 124 Gallant 162 Ratkowsky 395 Seber 457 Seber and Wild 458 or Ross 411 The first question is how to get suitable confidence intervals for the estimated parameters This is one of the main investigations when analyzing the output of data fitting Related questions are whether it is possible at all to identify parameters or how to eliminate redun dant ones as will be discussed in the subsequent sections For these and related modeling and simulation techniques see also the books of Walter and Pronzato 538 and of van den Bosch and van der Klauw 515 Another important question is experimental design where we want to create or improve existing experimental conditions The goals are to reduce the number of costly experiments to reduce error variances or to get identifiable parameters Typically initial values of differ ential equations or control functions e g for input feeds are to be adapted see e g Winer Brown and Michels 551 or Ryan 414 The standard tool to analyze the statistical properties of a dynamical model is the the evaluation of co
437. y be included INTEGER CONSTANT Definition of integer data either without index or with one or two dimensional index An index may be a variable or a constant number within an index set Also arithmetic integer expressions may be included TABLE lt identifier gt Assignment of constant real numbers to one or two dimensional array elements In subsequent lines one has to specify one or two indices followed by one real value per line in a free format starting at column 7 or later VARIABLE Declaration of variables with up to one index with respect to which automatic differ entiation is to be performed CONINT lt identifier gt Declaration of a piecewise constant interpolation function LININT lt identifier gt Declaration of a piecewise linear interpolation function SPLINE lt identifier gt Declaration of a spline interpolation function MACRO lt identifier gt Definition of a macro function an arbitrary set of PCOMP statements that define an auxiliary function to be inserted into subsequent function declaration blocks Macros are identified by a name that can be used in any right hand side of an assignment statement FUNCTION lt identifier gt Declaration of functions either with up to one index for which function and derivative values are to be evaluated The subsequent statements must assign a numerical value to the function identifier END End of the program It is recommended to follow the
438. yo p c The notation indicates that initial values can be fitted by EASY FIT Pes s without applying any special techniques If the initial time of a real dynamical process is not zero for example given by a to gt 0 then the model equations can be shifted easily back to zero by replacing each occurrence of t by t to in case of a non autonomous system It is important not to forget to shift also 37 25 20 15 10 Transferring Containers from Ship to Cargo Truck 0 02 04 06 0 8 12 14 16 18 2 Figure 2 19 Trajectories for Cargo Problem Transferring Containers from Ship to Cargo Truck 0 02 0 4 06 0 8 12 14 16 18 2 Figure 2 20 Control Functions for Cargo Problem 38 the measurement times t in the same way that is to replace all of them by t to for i 1 see Another possibility to start a dynamical system at to gt 0 is to define zero initial values for t 0 and to let the right hand side of the differential equation become zero from t 0 tot tg At to the true initial values and differential equations are then inserted It is necessary to declare to as a switching point see Section 2 4 3 to avoid non continuous transitions leading eventually to numerical instabilities A drawback of both approaches is that the final plots

e - Klaus Schittkowski

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