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User's Manual - Computation - Lawrence Livermore National

1. 0 0 0 0 0 0 0 6 1 0 1 1 1 1 1 1 1 2 3 0 0 0 0 0 0 Ni e 0 2 e 1 2 1 0 2 3 3 1 1 1 1 l 1 1 e 4 9 6 HYPRE StructStencil stencil int ndim 2 int size 5 int entry int offsets 2 0 0 1 0 1 0 0 1 0 1 Create the stencil object 1 StructStencilCreate ndim size amp stencil Set stencil entries for entry 0 entry lt size entry 1 2 6 HYPRE StructStencilSetElement stencil entry offsets entry I Thats it There is no assemble routine Figure 2 5 Code for setting up the stencil in Figure 12 CHAPTER 2 STRUCTURED GRID SYSTEM INTERFACE STRUCT HYPRE_StructMatrix A double values 36 int stencil_indices 2 0 3 int i HYPRE_StructMatrixCreate MPI_COMM_WORLD grid stencil amp A HYPRE_StructMatrixInitialize A for i 0 i lt 36 i 2 values i 4 0 values i 1 1 0 HYPRE StructMatrixSetBoxValues A ilower 0 iupper 0 2 Stencil indices values HYPRE StructMatrixSetBoxValues A ilower 1 iupper 1 2 Stencil indices values set boundary conditions HYPRE_StructMatrixAssemble A Figure 2 6 Code for setting up matrix values associated with stencil entries 0 and 3 as given by and Figure 2 4 SETTING UP THE STRUCT RIGHT HAND SIDE VECTOR 13 int ilower 2 3 1 int iupper 2 2 1 create matrix
2. In hypre they are one dimensional too It is usually obvious how to use them Just two functions in hypre GetRow and CoefficentAccess require a special type of array which Babel calls a SIDL array This has more structure than the arrays native to languages like 8 11 BUILDING HYPRE WITH THE BABEL INTERFACE 73 C or Fortran it accomodates reference counting and knows its own size for example The following examples show a way to declare read and destroy a SIDL array The GetRow and CoefficentAccess functions create their output arrays so there is no need to create your own For more information on these arrays read the Babel documentation at http www llnl gov CASC components C struct sidl int array row_js Struct sidl double array row data bHYPRE IJParCSRMatrix GetRow A i amp row size amp row js amp row data ex for k 0 k row size k col k sidl int array geti row js k data k sidl double array geti row data k Ssidl int array deleteRef row j ex Ssidl double array deleteRef row data ex Fortran integer 8 row js integer 8 row data call bHYPRE IJParCSRMatrix GetRow f A i row size row js 1 row data ex do k 1 row size call sidl int array get i f row js k 1 col k call sidl double array get f row data k 1 data k enddo call sidl int array deleteRef row j ex call sidl double array deleteRef row data ex 8 11 Building wit
3. soln ffsets solnValues where nodeIDList is a list of nodes in element block elemB1kID and solnO0ffsets i is the index pointing to the first location where the variables at node 7 is returned in solnValues 6 14 1 Solvers Available Only through the FEI While most of the solvers from the previous sections are available through the FEI interface there are number of additional solvers and preconditioners that are accessible only through the FEI These solvers are briefly described in this section see also the reference manual 6 14 FEI SOLVERS 59 Sequential and Parallel Solvers hypre currently has many iterative solvers There is also internally a version of the sequential SuperLU direct solver developed at U C Berkeley suitable to small problems may be up to the size of 10000 In the following we list some of these internal solvers 1 Additional Krylov solvers FGMRES TFQMR symmetric QMR 2 SuperLU direct solver sequential 3 SuperLU direct solver with iterative refinement sequential Parallel Preconditioners The performance of the Krylov solvers can be improved by clever selection of preconditioners Besides those mentioned previously in this chapter the following preconditioners are available via the LinearSystemCore interface 1 the modified version of MLI which requires the finite element substructure matrices to con struct the prolongation operators 2 parallel domain decomposition with inexact lo
4. F mpi comm MPI COMM WORLD call bHYPRE MPICommunicator CreateF f F mpi comm mpi comm ex call bHYPRE IJParCSRMatrix Create f mpi comm 8 7 Memory Management You will want to destroy whatever objects you create In C and Fortran you must do this explicitly Through the Babel based interface to hypre there are two ways to create something by calling a Create function or by calling a cast function You can destroy things with a Destroy or deleteRef function For every call of Create or cast there must be a call of Destroy or deleteRef Here is an example in C b bHYPRE IJParCSRVector Create mpi comm ilower iupper ex vb bHYPRE Vector cast b ex bHYPRE Vector deleteRef vb ex bHYPRE IJParCSRVector deleteRef b ex Here it is in Fortran call bHYPRE IJParCSRVector Create f mpi comm ilower iupper 1 b ex call bHYPRE Vector cast f b vb ex call bHYPRE IJParCSRVector deleteRef f vb ex call bHYPRE IJParCSRVector deleteRef f b ex What is actually going on here is memory management by reference counting Babel reference counts all its objects Reference counting means that the object contains an integer which counts the number of outside references to the object Babel will bump up the reference count by one when you call a Create or cast function Note that each of those functions normally is used to 70 CHAPTER 8 BABEL BASED INTERFACES assign an object to a variable In C and
5. factorization level to the value k 3 There are several methods of accomplishing this Internal to Euclid options are stored in a simple database that contains name value pairs Various of Euclid s internal private functions query this database to determine at runtime what action the user has requested If you enter the option stats 1 a report will be printed when Euclid s destructor is called this report lists among other statistics the options that were in effect during the factorization phase Method 1 By default Euclid always looks for a file titled database in the working directory If it finds such a file it opens it and attempts to parse it as a configuration file Configuration files should be formatted as follows gt cat database this is an optional comment level 3 Any line in a configuration file that contains a character in the first column is ignored All other lines should begin with an option name followed by one or more blanks followed by the option value Note that option names always begin with a character If you include an option name that is not recognized by Euclid no harm should ensue Method 2 To pass options on the command line call HYPRE EuclidSetParams HYPRE Solver solver int argc char argv where argc and argv carry the usual connotation main int argc char argv If your hypre application is called phoo you can then pass options on the command line
6. tories compilers used compile and load flags etc Executing configure results in the creation of platform specific files that are used when building the library The information may include such things as the system type being used for building and executing compilers being used libraries being searched option flags being set etc When all of the searching is done two files are left in the src directory config status contains information to recreate the current configuration and config log contains compiler messages which may help in debugging configure errors 57 58 CHAPTER 7 GENERAL INFORMATION Upon successful completion of configure the file config Makefile config is created from its template config Makefile config in and hypre is ready to be built Executing make with or without targets being specified in the src directory initiates compiling of all of the source code and building of the hypre library If any errors occur while compiling the user can edit the file config Makefile config directly then run make again without having to re run configure When building hypre without the install target the libraries and include files will be copied into the default directories src hypre lib and src hypre include respectively When building hypre using the install target the libraries and include files will be copied into the directories that the user specified in the options to configure e g prefix usr apps If none were spec
7. 18 It can be used both as a solver or as a preconditioner The user can choose between various different parallel coarsening techniques interpolation and relaxation schemes See for a detailed description of the coarsening algorithms interpolation and relaxation schemes as well as numerical results 6 8 1 Parameter Options Various BoomerAMG functions and options are mentioned below However for a complete listing and description of all available functions see the reference manual BoomerAMG s Create function differs from the synopsis in that it has only one parameter HYPRE BoomerAMGCreate HYPRE Solver solver It uses the communicator of the matrix A Coarsening can be set by the user using the function HYPRE_BoomerAMGSetCoarsenType Vari ous coarsening techniques are available 42 CHAPTER 6 SOLVERS AND PRECONDITIONERS the Cleary Luby Jones Plassman CLJP coarsening the Falgout coarsening which is a combination of CLJP and the classical RS coarsening algorithm default PMIS and HMIS coarsening algorithms which lead to coarsenings with lower complexities 5 as well as aggressive coarsening which can be applied to any of the coarsening techniques mentioned above and thus achieving much lower complexities and lower memory use 20 To use aggressive coarsening the user has to set the number of levels to which he wants to apply aggressive coarsening starting with the finest level via HYPRE_BoomerAMGSetAggNumLevels Since
8. FEI refers to a specific interface for black box finite element solvers originally developed in Sandia National Lab see 6 It differs from the rest of the conceptual interfaces in hypre in two important aspects it is written in C and it does not separate the construction of the linear system matrix from the solution process A complete description of Sandia s FEI implementation can be obtained by contacting Alan Williams at Sandia william sandia gov A simplified version of the FEI has been implemented at LLNL and is included in hypre More details about this implementation can be found in the header files of the FEI_mv fei base and FEI mv fei hypre directories 27 28 CHAPTER 4 FINITE ELEMENT INTERFACE 4 2 Brief Description of the Finite Element Interface Typically finite element codes contain data structures storing element connectivities element stiff ness matrices element loads boundary conditions nodal coordinates etc One of the purposes of the FEI is to assemble the global linear system in parallel based on such local element data We illustrate this in the rest of the section and refer to example 10 in the examples directory for more implementation details In hypre one creates an instance of the FEI as follows LLNL_FEI_Impl feiPtr new LLNL FEI Impl mpiComm Here mpiComm is an MPI communicator e g MPI COMM WORLD If Sandia s FEI package is to be used one needs to define a hypre solver object first
9. For more details about the ParaSails algorithm see 4 6 11 1 Parameter Settings The accuracy and cost of ParaSails are parameterized by the real thresh and integer nlevels pa rameters 0 lt thresh lt 1 0 nlevels Lower values of thresh and higher values of nlevels lead to more accurate but more expensive preconditioners More accurate preconditioners are also more expensive per iteration The default values are thresh 0 1 and nlevels 1 The parameters are set using HYPRE ParaSailsSetParams Mathematically given a symmetric matrix A the pattern of the approximate inverse is the pattern of A where A is a matrix that has been sparsified from A The sparsification is performed by dropping all entries in a symmetrically diagonally scaled A whose values are less than thresh in magnitude The parameter nlevel is equivalent to m 4 1 Filtering is a post thresholding procedure For more details about the algorithm see 4 The storage required for the ParaSails preconditioner depends on the parameters thresh and nlevels The default parameters often produce a preconditioner that can be stored in less than the space required to store the original matrix ParaSails does not need a large amount of intermediate storage in order to construct the preconditioner ParaSail s Create function differs from the synopsis in the followin way int HYPRE ParaSailsCreate MPI Comm comm HYPRE Solver solver int symmetry where comm is the MPI co
10. L PrecondSolver isa PCG conceptual interface class Figure 8 1 The ideas of the hypre object model IJVectorView StructVectorView lt es gt vs 4 v E L more than one view for any matrix class For vectors as well the Babel based interface is capable of multiple views but no more than one per class has been implemented For each matrix storage method there is a corresponding vector storage method When a matrix class contains a grid or not so does the corresponding vector class So there are four vector classes just as there are four matrix classes All vector classes inherit from a common interface to accomodate the common vector operations such as inner products sums and copies Here is an example of a full path through the inheritance tree In the SIDL file defining the Babel interface the StructVector class inherits from the StructVectorView and Vector interfaces Struct Vector View extends Matrix Vector View interface which extends the ProblemDefinition inter face The inheritance structure matters in use because sometimes you must cast an object up and down its inheritance hierarchy to provide an object of the right data type in a function call See section 8 10 Arrays Almost always when you pass an array through the Babel based interface you do it in the natural way you build and use the array type which is native to your language Babel calls this kind or array a raw array or rarray
11. LinearSystemCore solver HYPRE base create mpiComm FEI Implementation feiPtr FEI Implementation solver mpiComm rank where rank is the number of the master processor used only to identify which processor will produce the screen outputs The LinearSystemCore class is the part of the FEI which interfaces with the linear solver library It will be discussed later in Sections and 7 7 Local finite element information is passed to the FEI using several methods of the feiPtr object The first entity to be submitted is the field information A field has an identifier called fieldID and a rank or fieldSize number of degree of freedom For example a discretization of the Navier Stokes equations in 3D can consist of velocity vector having 3 degrees of freedom in every node vertex of the mesh and a scalar pressure variable which is constant over each element If these are the only variables and if we assign fieldIDs 7 and 8 to them respectively then the finite element field information can be set up by nFields 2 number of unknown fields fieldID new int nFields field identifiers fieldSize new int nFields vector dimension of each field velocity a 3D vector fieldID O T fieldSize 0 3 pressure a scalar function fieldID 1 8 fieldSize 1 1 feiPtr gt initFields nFields fieldSize fieldID Once the field information has been established we are ready to initialize an element
12. OTHER LANGUAGES 63 C parameter Fortran argument int i integer i double d double precision d int array integer array double array double precision array char string character string HYPRE_Type object integer 8 object HYPRE_Type object integer 8 object Table 7 1 Conversion from C parameters to Fortran arguments integer array N integer array 0 N 1 hypre objects can usually be declared as in the table because integer 8 usually corresponds to the length of a pointer However there may be some machines where this is not the case although we are not aware of any at this time On such machines the Fortran type for a hypre object should be an integer of the appropriate length This simple example illustrates the above information C prototype int HYPRE_IJMatrixSetValues HYPRE_IJMatrix matrix int nrows int ncols const int rows const int cols const double values The corresponding Fortran code for calling this routine is as follows integer 8 matrix integer nrows ncols MAX NCOLS integer rows MAX ROWS cols MAX COLS double precision values MAX COLS integer ierr call HYPRE IJMatrixSetValues matrix nrows ncols rows cols amp values ierr 64 CHAPTER 7 GENERAL INFORMATION Chapter 8 Babel based Interfaces 8 1 Introduction Much of hypre is accessible through a multi language interface built through the Babel tool This tool can connect to hypre fr
13. The third row of Figure is a list of data layouts or matrix vector storage schemes The relationship between linear solver and storage scheme is similar to that of the conceptual interface and linear solver Note that some of the interfaces in hypre currently only support one matrix vector storage scheme choice The conceptual interface the desired solvers and preconditioners and the matrix storage class must all be compatible 1 3 3 Writing your code As discussed in the previous section the following decisions should be made before writing any code 1 Choose a conceptual interface 2 Choose your desired solver strategy 3 Look up matrix requirements for each solver and preconditioner 4 Choose a matrix storage class that is compatible with your solvers and preconditioners and your conceptual interface Once the previous decisions have been made it is time to code your application to call hypre At this point reviewing the previously mentioned example codes provided with the hypre library may prove very helpful The example codes demonstrate the following general structure of the application calls to hypre 1 Build any necessary auxiliary structures for your chosen conceptual interface This includes e g the grid and stencil structures if you are using the structured grid interface 2 Build the matrix solution vector and right hand side vector through your chosen conceptual interface Each conceptual interface provides a seri
14. aggressive coarsening requires long range interpolation multipass interpolation is always used on levels with aggressive coarsening Various interpolation techniques can be set using HYPRE BoomerAMGSetInterpType the classical interpolation as defined in 18 default direct interpolation 20 standard interpolation multipass interpolation 20 an extended classical interpolation which is a long range interpolation and is recommended to be used with PMIS and HMIS coarsening for harder problems Jacobi interpolation 20 the classical interpolation modified for hyperbolic PDEs Jacobi interpolation is only use to improve certain interpolation operators and can be used with HYPRE BoomerAMGSetPostInterpType Since some of the interpolation operators might generate large stencils it is often possible and recommended to control complexity and truncate the interpo lation operators using HYPRE BoomerAMGSetTruncFactor and or HYPRE_BoomerAMGSetPMaxElmts or HYPRE_BoomerAMGSet JacobiTruncTheshold for Jacobi interpolation only Various relaxation techniques are available weighted Jacobi relaxation a hybrid Gauss Seidel Jacobi relaxation scheme a symmetric hybrid Gauss Seidel Jacobi relaxation scheme and hybrid block and Schwarz smoothers 23 ILU and approximate inverse smoothers 6 9 AMS 43 Point relaxation schemes can be set using HYPRE_BoomerAMGSetRelaxType or if one wants to specif ically
15. be set The final subsection lists the options and provides guidance as to the settings that in our experience will likely prove effective for minimizing execution time include HYPRE_parcsr_ls h HYPRE_Solver eu HYPRE_Solver pcg solver HYPRE_ParVector b x HYPRE_ParCSRMatrix A Instantiate the preconditioner HYPRE EuclidCreate comm Optionally use the following two calls to set runtime options 1 pass options from command line or string array HYPRE_EuclidSetParams eu argc argv 2 pass options from a configuration file HYPRE_EuclidSetParamsFromFile eu filename Set Euclid as the preconditioning method for some other solver using the function calls HYPRE_EuclidSetup 6 12 EUCLID 51 and HYPRE_EuclidSolve We assume that the pcg_solver has been properly initialized HYPRE_PCGSetPrecond pcg_solver HYPRE_PtrToSolverFcn HYPRE_EuclidSolve HYPRE_PtrToSolverFcn HYPRE_EuclidSetup Solve the system by calling the Setup and Solve methods for in this case the HYPRE_PCG solver We assume that A b and x have been properly initialized HYPRE_PCGSetup pcg_solver HYPRE_Matrix A HYPRE_Vector b HYPRE_Vector x HYPRE_PCGSolve pcg_solver HYPRE_Matrix parcsr_A HYPRE_Vector b HYPRE_Vector x Destroy the Euclid preconditioning object HYPRE_EuclidDestroy eu 6 12 2 Setting Options Examples For expositional purposes assume you wish to set the ILU k
16. bod be RGSS Ew 59 7 3 Testing the Library aoaaa ess 7 4 Linking to the Library 44 6 a amp dope AR eis eR dE ee ITO 7 5 Error Flags co a a be we ae Ae ee A ee a Ge ee A a 7 6 Bug Reporting and General Support 7 7 Using in External FEI Implementations 7 8 Calling from Other Languages Babel based Interfaces Introduction ok Re eoe eee RE ek Sa oe ee NINOS Wie 6 2 Interfaces Similar 2e wes doe ck ee ee ROO Y RR m oe e eg wk eee 8 3 Interfaces Are Different 8 4 Names and Conventions gt so cso 0 8 5 Parameters and Error 67 amp 6 MPI Communicator ue uas see ane ee aed a a RA WORD Roe E RO Xd Ue i 68 8 7 Memory Management OS SOAS aou x uuo UE ded MERE and e Ee A ROCA ELDER X 70 89 Object Structure S10 Arrays a moe cene cx we dr E Rub aub LE dea x d Foie E EAE ORGS CONTENTS 8 11 Building HYPRE with the Babel Interface 8 11 1 Building with Python Using the Babel Interface ii CONTENTS Chapter 1 Introduction This manual describes hypre a software library of high performance preconditioners and solvers for the solution of large sparse
17. e Structured Grid System Interface Struct This interface is appropriate for applica tions whose grids consist of unions of logically rectangular grids with a fixed stencil pattern of nonzeros at each grid point This interface supports only a single unknown per grid point See Chapter 2 for details 4 CHAPTER 1 INTRODUCTION Linear System Interfaces HB VER Linear Solvers ot Xxx X Data Layout Figure 1 1 Graphic illustrating the notion of conceptual interfaces e Semi Structured Grid System Interface SStruct This interface is appropriate for applications whose grids are mostly structured but with some unstructured features Exam ples include block structured grids composite grids in structured adaptive mesh refinement AMR applications and overset grids This interface supports multiple unknowns per cell See Chapter 3 for details e Finite Element Interface FEI This is appropriate for users who form their linear sys tems from a finite element discretization The interface mirrors typical finite element data structures including element stiffness matrices Though this interface is provided in hypre its definition was determined elsewhere please email to Alan Williams william sandia gov for more information See Chapter 4 for details e Linear Algebraic System Interface IJ This is the traditional linear algebraic inter face It can be used as a last resort by users for whom the other gr
18. for i 0 i lt nElems i feiPtr gt sumInElem elemBlkID elemID elemConn i elemStiff i elemLoads i elemFormat To complete the assembling of the global stiffness matrix and the corresponding right hand side a signal is sent to the FEI via feiPtr gt loadComplete Chapter 5 Linear Algebraic System Interface IJ The IJ interface described in this chapter is the lowest common denominator for specifying linear systems in hypre This interface provides access to general sparse matrix solvers in hypre not to the specialized solvers that require more problem information 5 1 IJ Matrix Interface As with the other interfaces in hypre the IJ interface expects to get data in distributed form because this is the only scalable approach for assembling matrices on thousands of processes Matrices are assumed to be distributed by blocks of rows as follows Ao A 7 5 1 Ap 1 In the above example the matrix is distributed accross the P processes 0 1 P 1 by blocks of rows Each submatrix A is owned by a single process and its first and last row numbers are given by the global indices ilower and iupper in the Create call below The following example code illustrates the basic usage of the IJ interface for building matrices MPI_Comm comm HYPRE_IJMatrix ij_matrix HYPRE_ParCSRMatrix parcsr_matrix int ilower iupper int jlower jupper int nrows 31 32 CHAPTER 5 LINEAR ALGEBRAIC SYSTEM IN
19. in the mesh with each row having two nonzero entries 1 and 1 in the columns corresponding to the vertices composing the edge The sign is determined based on the orientation of the edge We require that G includes all interior and boundary edges and vertices 2 The representations of the constant vector fields 1 0 0 0 1 0 and 0 0 1 in the V basis given as three vectors Gz Gy and Gz Note that since no boundary conditions are imposed on the above vectors can be computed as Gr Gy Gy and Gz where y and z are vectors representing the coordinates of the vertices of the mesh In addition to the above quantities AMS can utilize the following optional information 3 The Poisson matrices and Ag corresponding to assembling of the forms a Vu Vv and B Vu Vv using standard linear finite elements on the same mesh Internally AMS proceeds with the construction of the following additional objects e Ag a matrix associated with the mass term which is either or the Poisson matrix Ag if given e II the matrix representation of the interpolation operator from vector linear to edge finite elements e Ay a matrix associated with the stiffness term which is either II AII or a block diagonal matrix with diagonal blocks A if given e and efficient AMG solvers for Ag and The solution procedure then is a three level method using smoothing in the origina
20. lib name with lapack lib dirs path to lapack 7 3 TESTING THE LIBRARY 59 The output from configure is several pages long It reports the system type being used for building and executing compilers being used libraries being searched option flags being set etc 7 2 2 Make Targets The make step in building hypre is where the compiling loading and creation of libraries occurs Make has several options that are called targets These include help prints the details of each target all default target in all directories compile the entire library does NOT rebuild documentation clean deletes all files from the current directory that are created by building the library distclean deletes all files from the current directory that are created by configuring or building the library install compile the source code build the library and copy executables libraries etc to the appropriate directories for user access uninstall deletes all files that the install target created tags runs etags to create a tags table file is named TAGS and is saved in the top level directory test depends on the all target to be completed removes existing temporary installation directories creates temporary installation directories copies all libHYPRE and h files to the temporary locations builds the test drivers linking to the temporary locations to simulate how application codes will link to HYPRE 7 3 Testing the Library The examples subdire
21. machines SIAM J Sci Comput 21 5 1823 1834 2000 Special issue on the Fifth Copper Mountain Conference on Iterative Methods Also available as LLNL technical report UCRL JC 130720 E Chow A priori sparsity patterns for parallel sparse approximate inverse preconditioners SIAM J Sci Comput 21 1804 1822 2000 H De Sterck U M Yang and J Heys Reducing complexity in parallel algebraic multigrid preconditioners SIAM Journal on Matrix Analysis and Applications 27 1019 1039 2006 Also available as LLNL technical report UCRL JRNL 206780 R L Clay et al An annotated reference guide to the Finite Element Interface FEI specifica tion Version 1 0 Technical Report SAND99 8229 Sandia National Laboratories Livermore CA 1999 R D Falgout and J E Jones Multigrid on massively parallel architectures In E Dick K Riemslagh and J Vierendeels editors Multigrid Methods VI volume 14 of Lecture Notes in Computational Science and Engineering pages 101 107 Berlin 2000 Springer Proc of the Sixth European Multigrid Conference held in Gent Belgium September 27 30 1999 Also available as LLNL technical report UCRL JC 133948 V E Henson and U M Yang BoomerAMG a parallel algebraic multigrid solver and pre conditioner Applied Numerical Mathematics 41 5 155 177 2002 Also available as LLNL technical report UCRL JC 141495 Hiptmair and J Xu Nodal auxiliary space preconditioning in H curl di
22. number of coefficients in the row that couple the diagonal unknown to 0ffd unknowns in other processor domains HYPRE IJMatrixSetRowSizes ij matrix sizes HYPRE IJMatrixSetDiag 0ffdSizes matrix diag sizes offdiag sizes Once the matrix has been assembled the sparsity pattern cannot be altered without completely destroying the matrix object and starting from scratch However one can modify the matrix values of an already assembled matrix To do this first call the Initialize routine to re initialize the matrix then set or add to values as before and call the Assemble routine to re assemble before using the matrix Re initialization and re assembly are very cheap essentially a no op in the current implementation of the code 5 2 IJ Vector Interface The following example code illustrates the basic usage of the IJ interface for building vectors MPI Comm comm HYPRE IJVector ij vector HYPRE ParVector par vector int jlower jupper int nvalues int indices double values HYPRE IJVectorCreate comm jlower jupper amp ij vector HYPRE IJVectorSet bjectType ij vector HYPRE PARCSR HYPRE IJVectorInitialize ij vector set vector values HYPRE IJVectorSetValues ij vector nvalues indices values HYPRE IJVectorAssemble ij vector HYPRE IJVectorGet bject ij vector void amp par vector The Create routine creates an empty vector object that lives on the comm communicator This is a coll
23. suited for block structured grid problems There are five basic steps involved in setting up the linear system to be solved set up the grid set up the stencils 1 2 3 set up the graph 4 set up the matrix 5 set up the right hand side vector 17 18 CHAPTER 3 SEMI STRUCTURED GRID SYSTEM INTERFACE SSTRUCT Figure 3 1 Grid variables in hypre are referenced by the abstract cell centered index to the left and down in 2D analogously in 3D In the figure index i j is used to reference the variables in black The variables in grey although contained in the pictured cell are not referenced by the i j index 3 1 Block Structured Grids In this section we describe how to use the SStruct interface to define block structured grid prob lems We will do this primarily by example paying particular attention to the construction of stencils and the use of the GridSetNeighborbox interface routine Consider the solution of the diffusion equation V DVu ou f 3 1 on the block structured grid in Figure where D is a scalar diffusion coefficient and o gt 0 The discretization introduces three different types of variables cell centered z face and y face The three discretization stencils that couple these variables are also given in the figure The information in this figure is essentially all that is needed to describe the nonzero structure of the linear system we wish to solve The grid in F igure
24. the system into a 3 x 3 block matrix A A N 11 A1 ieee Ag Ag A NT D 0 21 22 The reduced system has the form Ai 5 Aig 21 b Ap be which is symmetric positive definite SPD if the original matrix is PD In addition is easy to compute There are three slide surface reduction algorithms in hypre The first follows the matrix formu lation in this section The second is similar except that it replaces the eliminated slave equations with identity rows so that the degree of freedom at each node is preserved This is essential for certain block algorithms such as the smoothed aggregation multilevel preconditioners The third is similar to the second except that it is more general and can be applied to problems with intersecting slide surfaces sequential only for intersecting slide surfaces Other Features To improve the efficiency of the hypre solvers a few other features have been incorporated We list a few of these features below 1 Preconditioner reuse For multiple linear solves with matrices that are slightly perturbed from each other oftentimes the use of the same preconditioners can save preconditioner setup times but suffer little convergence rate degradation 2 Projection methods For multiple solves that use the same matrix previous solution vectors can sometimes be used to give a better initial guess for subsequent solves Two projection schemes have been implemented in hypre
25. to hypre functions In particular a value of zero means that all hypre functions up to and including the current one have completed successfully This new error flags system is being implemented throughout the library but currently there are still functions that do not support it The error flag value is a combination of one or a few of the following error codes 1 HYPRE ERROR GENERIC describes a generic error 2 HYPRE ERROR MEMORY hypre was unable to allocate memory 3 ERROR error in one of the arguments of a hypre function 4 HYPRE ERROR CONV a hypre solver did not converge as expected One can use the HYPRE CheckError function to determine exactly which errors have occurred call some HYPRE functions hypre ierr HYPRE Function check if the previously called hypre functions returned error s if hypre ierr check if the error with code HYPRE ERROR CODE has occurred if HYPRE CheckError hypre ierr HYPRE ERROR CODE The corresponding FORTRAN code is C header file with hypre error codes include HYPRE error f h C call some HYPRE functions 7 6 BUG REPORTING AND GENERAL SUPPORT 61 call HYPRE_Function hypre ierr C check if the previously called hypre functions returned error s if hypre ierr ne 0 then C check if the error with code HYPRE ERROR CODE has occurred HYPRE CheckError hypre ierr HYPRE ERROR CODE check if check ne 0 then The global error flag can als
26. will need write access to your Python s site packages directory When you install pyMPI it creates another instance of Python so this should not be a problem You must configure and build hypre with shared libraries because Python requires them And specify what Python you are building for files will be written into its site packages directory For example configure with babel enable shared enable python pyMPI make Finally you need to set two environment variables LD_LIBRARY_PATH must include the path to the hypre shared libraries e g src hypre lib SIDL DLL PATH must include the path to an scl file generated for the Python interface e g Src babel bHYPREClient P libbHYPRE scl If you are running multiple processes these environment variables may need to be set in a dotfile so all processes will use the same values See the examples directory for an example of how to write a Python program which uses hypre Bibliography 1 S Ashby and R D Falgout A parallel multigrid preconditioned conjugate gradient al gorithm for groundwater flow simulations Nuclear Science and Engineering 124 1 145 159 September 1996 Also available as LLNL Technical Report UCRL JC 122359 A H Baker R D Falgout and U M Yang An assumed partition algorithm for determining processor inter communication Parallel Computing 32 394 414 2006 P N Brown R D Falgout and J E Jones Semicoarsening multigrid on distributed memory
27. 6 ierr amg solver SetDoubleParameter Tolerance 1e 7 Python Babel ierr solver SetIntParameter CoarsenType 6 solver SetDoubleParameter Tolerance 1e 7 8 6 MPI Communicator In the C only hypre interface and most normal MPI usage one often needs an MPI communicator of type MPI_Comm What an MPI_Comm really is depends on the language and the MPI implementation But the Babel interface is supposed to be fundamentally independent of languages and imple mentations So the MPI communicator is wrapped in a special bHYPRE MPICommunicator object This not an MPI Comm object is what you pass to all the Babel interface functions which need an MPI communicator Thus the language dependence of MPI is isolated in the function which creates the bHYPRE MPICommunicator object Here are examples of how to use this function C Babel bHYPRE MPICommunicator mpi comm MPI Comm mpicommworld MPI COMM WORLD MPI Comm C mpi comm amp mpicommworld mpi_comm bHYPRE MPICommunicator CreateC C mpi comm ex parcsr_A bHYPRE IJParCSRMatrix Create mpi comm 8 7 MEMORY MANAGEMENT 69 C 4 Babel using namespace ucxx bHYPRE MPICommunicator mpi comm MPI Comm mpicommworld MPI COMM WORLD MPI Comm C mpi comm amp mpicommworld mpi comm MPICommunicator CreateC C mpi comm parcsr_A IJParCSRMatrix Create mpi comm Fortran Babel integer 8 mpi comm integer 8 F mpi comm
28. A conjugate projection for SPD matrices and minimal residual projection for both SPD and non SPD matrices 3 The sparsity pattern of the matrix is in general not destroyed after it has been loaded to an hypre matrix But if the matrix is not to be reused an option is provided to clean up this pattern matrix to conserve memory usage Chapter 7 General Information 7 1 Getting the Source Code The hypre distribution tar file is available from the Software link of the hypre web page Wwww l1lnl gov CASC hypre The hypre Software distribution page allows access to the tar files of the latest and previous general and beta distributions as well as documentation 7 2 Building the Library After unpacking the hypre tar file the source code will be in the src sub directory of a directory named hypre VERSION where VERSION is the current version number e g hypre 1 8 4 with a b appended for a beta release Move to the src sub directory to build hypre for the host platform The simplest method is to configure compile and install the libraries in hypre lib and hypre include directories which is accomplished by configure make NOTE when executing on an IBM platform configure must be executed under the nopoe script nopoe configure option lt option gt to force a single task to be run on the log in node There are many options to configure and make to customize such things as installation direc
29. B 2 is defined in terms of five separate logically rectangular parts as shown in Figure 3 and each part is given a unique label between 0 and 4 Each part consists of a single box with lower index 1 1 and upper index 4 4 see Section 2 1 and the grid data is distributed on five processes such that data associated with part p lives on process p Note that in general parts may be composed out of arbitrary unions of boxes and indices may consist of non positive integers see Figure 2 2 Also note that the SStruct interface expects a domain based data distribution by boxes but the actual distribution is determined by the user and simply described in parallel through the interface As with the Struct interface each process describes that portion of the grid that it owns one box at a time F igure B 4 shows the code for setting up the grid on process 3 the code for the other processes is similar The icons at the top of the figure illustrate the result of the numbered lines of code Process 3 needs to describe the data pictured in the bottom right of the figure That is it needs to describe part 3 plus some additional neighbor information that ties part 3 together 3 1 BLOCK STRUCTURED GRIDS 19 pou ko Aec g i e VES or X 9 a Y oec Figure 3 2 Example of a block structured grid with five logically rectangular blocks and three variab
30. C only interface but usually they are the same Most hypre objects need to be supplied with an MPI communicator The Babel interface has a special class for the MPI communicator for details see section 8 5 Parameters and Error Flags Most hypre objects can be modified by setting parameters In the C only interface there is a separate function to set each parameter The Babel based interface has just a few parameter setting functions for each object The parameters are identified by their names as strings See below for some examples Most hypre functions return error flags as discussed elsewhere in this manual The following examples simply show how they are returned not how to handle them C only HYPRE Solver solver int ierr ierr HYPRE BoomerAMGSetCoarsenType amg solver 6 ierr HYPRE BoomerAMGSetTol amg solver 1e 7 C Babel bHYPRE BoomerAMG amg solver 68 CHAPTER 8 BABEL BASED INTERFACES int ierr ierr bHYPRE_BoomerAMG_SetIntParameter amg_solver CoarsenType 6 ex ierr bHYPRE_BoomerAMG_SetDoubleParameter amg_solver Tolerance 1e 7 ex Fortran Babel integer 8 amg_solver integer ierr call bHYPRE_BoomerAMG_SetIntParameter_f 1 amg_solver CoarsenType 6 ierr ex call bHYPRE_BoomerAMG_SetDoubleParameter_f 1 amg_solver Tolerance tol ierr ex C4 4 Babel using namespace ucxx bHYPRE BoomerAMG amg_solver ierr amg solver SetIntParameter CoarsenType
31. Create MPI_COMM_WORLD ndim amp grid Set grid extents for the first box HYPRE StructGridSetExtents grid ilower 0 iupper 0 Set grid extents for the second box HYPRE StructGridSetExtents grid ilower 1 iupper 1 Assemble the grid HYPRE StructGridAssemble grid Figure 2 3 Code on process 0 for setting up the grid in Figure 2 1 10 CHAPTER 2 STRUCTURED GRID SYSTEM INTERFACE STRUCT 8 0 0 0 0 0 M S e 1 0 2 e 3 gt 0 1 2 4 0 1 1 1 e Figure 2 4 Representation of the 5 point discretization stencil for the example problem 2 2 Setting Up the Struct Stencil The geometry of the discretization stencil is described by an array of indexes each representing a relative offset from any given gridpoint on the grid For example the geometry of the 5 pt stencil for the example problem being considered can be represented by the list of index offsets shown in Figure 2 4 Here the 0 0 entry represents the center coefficient and is the Oth stencil entry The 0 1 entry represents the south coefficient and is the 3rd stencil entry And so on On process 0 or 1 the code in F igure 2 5 will set up the stencil in F igure 2 4 The stencil must be the same on all processes The Create routine creates an empty 2D 5 pt stencil object The SetElement routine defines the geometry of the stencil and assigns t
32. Fortran it is up to you do decrement the reference count when that variable is no longer needed With Babel reference counting is automatic in C so you will not normally need to do any memory management yourself If you are interested in doing more with reference counting see the Babel users manual for more information In some cases you may find Babel s memory tools useful e g if you copy pointers to Babel objects 8 8 Casting If your code is written in C or Fortran you will occasionally need to call a Babel cast function This section tells you why and how Functions are generally written in as much generality as possible For example the PCG algorithm works the same for any kind of matrix and vector as long as it can multiply a matrix and vector compute an inner product and so forth So two of its arguments are of type bHYPRE Vector But when you create a vector you have to decide how it s going to be stored So you have to declare it as a more specific data type e g DHYPRE_StructVector To apply the PCG solver to that vector in C or Fortran you have to call a cast function to change its data type In C and Python you do not normally need to call a cast function Be careful to call a deleteRef or Destroy function to correspond to every cast function as discussed in section Here are examples of casting in C and Fortran bHYPRE_StructVector b_S bHYPRE_StructVector_Create bHYPRE_Vector b bHYPRE_Vector__cas
33. SSetDiscreteGradient solver G In addition to G we need one additional piece of information in order to construct the solver The user has the option to choose either the coordinates of the vertices in the mesh or the repre sentations of the constant vector fields in the edge element basis In both cases three hypre parallel vectors should be provided For 2D problems the user can set the third vector to NULL The corresponding function calls read HYPRE_AMSSetCoordinateVectors solver x y Z or HYPRE_AMSSetEdgeConstantVectors solver one_zero_zero zero_one_zero zero_zero_one The vectors one_zero_zero zero_one_zero and zero_zero_one above correspond to the constant vector fields 1 0 0 0 1 0 and 0 0 1 The remaining solver parameters are optional For example the user can choose a different cycle type by calling HYPRE AMSSetCycleType solver cycle type default value 1 The available cycle types in AMS are e cycle type 1 multiplicative solver 01210 e cycle type 2 additive solver 0 1 2 e cycle type 3 multiplicative solver 02120 e cycle type 4 additive solver 010 4 2 e cycle type 5 multiplicative solver 0102010 e cycle type 6 additive solver 1 4 020 e cycle type 7 multiplicative solver 0201020 e cycle type 8 additive solver 010 020 46 CHAPTER 6 SOLVERS AND PRECONDITIONERS Here we use the following convention for the three subspace correction methods 0 refers to smo
34. TERFACE IJ int ncols int rows int cols double values HYPRE IJMatrixCreate comm ilower iupper jlower jupper amp ij matrix HYPRE IJMatrixSetObjectType ij matrix HYPRE PARCSR HYPRE IJMatrixInitialize ij matrix set matrix coefficients HYPRE IJMatrixSetValues ij matrix nrows ncols rows cols values add to matrix cofficients if desired HYPRE IJMatrixAddToValues ij matrix nrows ncols rows cols values HYPRE IJMatrixAssemble ij matrix HYPRE IJMatrixGet 0bject ij matrix void amp parcsr matrix The Create routine creates an empty matrix object that lives on the comm communicator This is a collective call i e must be called on all processes from a common synchronization point with each process passing its own row extents ilower and iupper The row partitioning must be contiguous i e iupper for process i must equal ilower 1 for process i 1 Note that this allows matrices to have 0 or 1 indexing The parameters jlower and jupper define a column partitioning and should match ilower and iupper when solving square linear systems See the Reference Manual for more information The SetObjectType routine sets the underlying matrix object type to HYPRE_PARCSR this is the only object type currently supported The Initialize routine indicates that the matrix coefficients or values are ready to be set This routine may or may not involve the allocation of memory for
35. User s Manual Software Version 2 0 0 Date 2006 12 15 MGM perfanmame precanmitiamens Center for Applied Scientific Computing Lawrence Livermore National Laboratory Copyright c 2006 The Regents of the University of California Produced at the Lawrence Liv ermore National Laboratory Written by the HYPRE team UCRL CODE 222953 All rights reserved This file is part of see http www llnl gov CASC hypre Please see the COPY RIGHT and LICENSE file for the copyright notice disclaimer contact information and the GNU Lesser General Public License HYPRE is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation version 2 1 dated February 1999 HYPRE is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the IMPLIED WARRANTY OF MERCHANTABILITY or FITNESS FOR A PAR TICULAR PURPOSE See the terms and conditions of the GNU General Public License for more details You should have received a copy of the GNU Lesser General Public License along with this program if not write to the Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Contents 1 Introduction 1 1 Overview of Features 2 2222 ee 1 2 Getting More 1 3 How to get started 1 3 1 Installing hyp
36. ain is assigned a unique global identifier The shared node list on each processor contains a subset of the global node list corresponding to the local nodes that are shared with the other processors The syntax for setting up the shared nodes is feiPtr gt initSharedNodes nShared sharedIDs sharedLengs sharedProcs This completes the initialization phase and a completion signal is sent to the FEI via feiPtr gt initComplete Next we begin the load phase The first entity for loading is the nodal boundary conditions Here we need to specify the number of boundary equations and the boundary values given by alpha beta and gamma Depending on whether the boundary conditions are Dirichlet Neumann or mixed the three values should be passed into the FEI accordingly feiPtr gt loadNodeBCs nBCs BCEqn fieldID alpha beta gamma The element stiffness matrices are to be loaded in the next step We need to specify the element number i the element block to which element i belongs the element connectivity information the element load and the element matrix format The element connectivity specifies a set of 8 node global IDs for hexahedral elements and the element load is the load or force for each degree of freedom The element format specifies how the equations are arranged similar to the interleaving scheme mentioned above The calling sequence for loading element stiffness matrices is 30 CHAPTER 4 FINITE ELEMENT INTERFACE
37. an tailor the installation to match their computing system Options include debug and optimized modes the ability to change required libraries such as MPI and BLAS a sequential mode and modes enabling threads for certain solvers On most systems however hypre can be built by simply typing configure followed by make e Interfaces in multiple languages provide greater flexibility for applications hypre is written in C with the exception of the FEI interface which is written in C and utilizes Babel to provide interfaces for users of Fortran 77 Fortran 90 C Python and Java For more information on Babel see http www 11n1 gov CASC components babel htm1 1 2 Getting More Information This user s manual consists of chapters describing each conceptual interface a chapter detailing the various linear solver options available and detailed installation information In addition to this manual a number of other information sources for hypre are available e Reference Manual The reference manual comprehensively lists all of the interface and solver functions available in hypre The reference manual is ideal for determining the various options available for a particular solver or for viewing the functions provided to describe a problem for a particular interface e Example Problems A suite of example problems is provided with the hypre installation These examples reside in the examples subdirectory and demonstrate various features of
38. and set interior coefficients implement boundary conditions for i 0 i lt 12 i values i 0 0 i 3 HYPRE_StructMatrixSetBoxValues A ilower iupper 1 amp i values complete implementation of boundary conditions Figure 2 7 Code for adjusting boundary conditions along the lower grid boundary in Figure owns The Assemble routine is a collective call and finalizes the matrix assembly making the matrix ready to use Matrix coefficients that reach outside of the boundary should be set to zero For efficiency reasons hypre does not do this automatically The most natural time to insure this is when the boundary conditions are being set and this is most naturally done after the coefficients on the grid s interior have been set For example during the implementation of the Dirichlet boundary condition on the lower boundary of the grid in Figure the south coefficient must be set to zero To do this on process 0 the code in Figure could be used 2 4 Setting Up the Struct Right Hand Side Vector The right hand side vector is set up similarly to the matrix set up described in Section above The main difference is that there is no stencil note that a stencil currently does appear in the interface but this will eventually be removed On process 0 the code in F igure 2 8 will set up the right hand side vector values The Create routine creates an empty vector object The Initia
39. ar problems fastest solution time is typically obtained with levels 4 through 8 For 3D problems fastest solution time is typically obtained with level 1 bj Use Block Jacobi ILU preconditioning instead of PILU Default 0 false Guidance if sub domains contain relatively few nodes less than 1 000 or the problem is not well partitioned Block Jacobi ILU may give faster solution time than PILU eu stats When Euclid s destructor is called a summary of runtime settings and timing information is printed to stdout Default 0 false The timing marks in the report are the maximum over all processors in the MPI communicator eu mem When Euclid s destructor is called a summary of Euclid s memory usage is printed to std out Default 0 false The statistics are for the processor whose rank in MPI COMM WORLD is 0 printTestData This option is used in our autotest procedures and should not normally be invoked by users The following options are partially implemented but not yet fully functional i e don t use them until further notice sparseA float Drop tolerance for ILU k factorization Default 0 no dropping Entries are treated as zero if their absolute value is less than sparseA max where max is the largest absolute value of any entry in the row Guidance try this in conjunction with rowScale 6 13 PILUT PARALLEL INCOMPLETE FACTORIZATION 53 CAUTION If the coefficient matrix A is symmetric this sett
40. ations see Figure IA overset grids In addition it supports more general PDEs than the Struct interface by allowing multiple variables system PDEs and multiple variable types e g cell centered face centered etc The interface provides access to data structures and linear solvers in hypre that are designed for semi structured grid problems but also to the most general data structures and solvers These latter solvers are usually provided via the FEI or IJ interfaces described in Chapters 4 and 5 The SStruct grid is composed out of a number of structured grid parts where the physical inter relationship between the parts is arbitrary Each part is constructed out of two basic components boxes see Figure and variables Variables represent the actual unknown quantities in the grid and are associated with the box indices in a variety of ways depending on their types In hypre variables may be cell centered node centered face centered or edge centered Face centered variables are split into x face y face and z face and edge centered variables are split into x edge y edge and z edge See Figure 3 1 for an illustration in 2D The SStruct interface uses a graph to allow nearly arbitrary relationships between part data The graph is constructed from stencils plus some additional data coupling information set by the GraphAddEntries routine Another method for relating part data is the GridSetNeighborbox routine which is particularly
41. be called from a variety of languages This is currently provided through two basic mechanisms e The Babel interfaces described in Chapter 8 utilize the Babel tool to provide the most exten sive language support note that the hypre library is moving towards using the Babel tool as its primary means of getting language interoperability e A Fortran interface is manually supported to mirror the native C interface used throughout most of this manual We describe this interface next Typically the Fortran subroutine name is the same as the C name unless it is longer than 31 characters In these situations the name is condensed to 31 characters usually by simple truncation For now users should look at the Fortran test drivers f codes in the test directory for the correct condensed names In the future this aspect of the interface conversion will be made consistent and straightforward The Fortran subroutine argument list is always the same as the corresponding C routine except that the error return code ierr is always last Conversion from C parameter types to Fortran argument type is summarized in Table Array arguments in hypre are always of type int or double and the corresponding Fortran types are simply integer or double precision arrays Note that the Fortran arrays may be indexed in any manner For example an integer array of length N may be declared in fortran as either of the following 7 8 CALLING HYPRE FROM
42. bels between 0 and 8 and their offsets are described relative to the center of the stencil This process is illustrated in FigureB 5 Nine calls are made to the routine HYPRE SStructStencilSetEntry As an example the call that describes stencil entry 5 in the figure is given the entry number 5 the offset 1 0 and the identifier for the x face variable the variable to which this entry couples Recall from FigureB 1 the convention used for referencing variables of different types The geometry description uses the same convention but with indices numbered relative to the referencing index 0 0 for the stencil s center Figure 3 6 shows the code for setting up the graph With the above we now have a complete description of the nonzero structure for the matrix The matrix coefficients are then easily set in a manner similar to what is described in Section 2 3 using routines MatrixSetValues and MatrixSetBoxValues in the SStruct interface As before 22 CHAPTER 3 SEMI STRUCTURED GRID SYSTEM INTERFACE SSTRUCT 0 0 0 1 0 1 1 1 e o B 01 4 2 3 00 4 5 4 0 1 1 0 10 gt 1 6 0 0 gt 7 1 1 gt 0 1 8 lt gt 01 gt Figure 3 5 Assignment of labels and geometries to the y face stencil in Figure 3 2 Stencil offsets are described relative to the 0 0 index for the center of the stencil there are also AddTo variant
43. block An element block is characterized by the block identifier the number of elements the number of nodes per element the nodal fields and the element fields fields that have been defined previously Suppose we use 1000 hexahedral elements in the element block 0 the setup consists of 4 2 A BRIEF DESCRIPTION OF THE FINITE ELEMENT INTERFACE 29 elemBlkID 0 identifier for a block of elements nElems 1000 number of elements in the block elemNNodes 8 number of nodes per element nodal based field for the velocity nodeNFields 1 nodeFieldIDs new nodeNFields nodeFieldIDs O fieldID 0 element based field for the pressure elemNFields 1 elemFieldIDs new elemNFields elemFieldIDs 0 fieldID 1 feiPtr gt initElemBlock elemBlkID nElems elemNNodes nodeNFields nodeFieldIDs elemNFields elemFieldIDs 0 The last argument above specifies how the dependent variables are arranged in the element matrices A value of 0 indicates that each variable is to be arranged in a separate block as opposed to interleaving In a parallel environment each processor has one or more element blocks Unless the element blocks are all disjoint some of them share a common set of nodes on the subdomain boundaries To facilitate setting up interprocessor communications shared nodes between subdomains on different processors are to be identified and sent to the FEI Hence each node in the whole dom
44. cal solves DDllut 3 least squares polynomial preconditioner 4 2 x 2 block preconditioner and 5 2 x 2 Uzawa preconditioner Some of these preconditioners can be tuned by a number of internal parameters modifiable by users A description of these parameters is given in the reference manual Matrix Reduction For some structural mechanics problems with multi point constraints the discretization matrix is indefinite eigenvalues lie in both sides of the imaginary axis Indefinite matrices are much more different to solve than definite matrices Methods have been developed to reduce these indefinite matrices to definite matrices Two matrix reduction algorithms have been implemented in hypre as presented in the following subsections Schur Complement Reduction The incoming linear system of equations is assumed to be in the form D B xy _ by BT 0 29 B where D is a diagonal matrix After Schur complement reduction is applied the resulting linear system becomes b B D 1by 56 CHAPTER 6 SOLVERS AND PRECONDITIONERS Slide Surface Reduction With the presence of slide surfaces the matrix is in the same form as in the case of Schur complement reduction Here A represents the relationship between the master slave and other degrees of freedom The matrix block B70 corresponds to the constraint equations The goal of reduction is to eliminate the constraints As proposed by Manteuffel the trick is to re order
45. ceptual interface An important decision to make before writing any code is to choose an appropriate conceptual interface These conceptual interfaces are intended to represent the way that applications developers naturally think of their linear problem and to provide natural interfaces for them to pass the data that defines their linear system into hypre Essentially these conceptual interfaces can be considered convenient utilities for helping a user build a matrix data structure for hypre solvers and preconditioners The top row of Figure illustrates a number of conceptual interfaces Generally the conceptual interfaces are denoted by different types of computational grids but other application features might also be used such as geometrical information For example applications that use structured grids such as in the left most interface in the Figure 1 1 typically view their linear problems in terms of stencils and grids On the other hand applications that use unstructured grids and finite elements typically view their linear problems in terms of elements and element stiffness matrices Finally the right most interface is the standard linear algebraic matrix rows columns way of viewing the linear problem The hypre library currently supports four conceptual interfaces and typically the appropriate choice for a given problem is fairly obvious e g a structured grid interface is clearly inappropriate for an unstructured grid application
46. ctory contains several codes that can be used to test the newly created hypre library To create the executable versions move into the examples subdirectory enter make then execute the codes as described in the initial comments section of each source code 60 CHAPTER 7 GENERAL INFORMATION 7 4 Linking to the Library An application code linking with hypre must be compiled with I PREFIX include and linked with L PREFIX lib 1HYPRE where PREFIX is the directory where hypre is installed default is hypre or as defined by the configure option prefix PREFIX As noted above if hypre was built as a shared library the user MUST have its location defined in the environment variable LD_LIBRARY_PATH We strongly recommend that users link with 11bHYPRE a rather than specifying each of the sep arate hypre libraries LibHYPRE_krylov a libHYPRE parcsr mv a etc By using libHYPRE a the user is guaranteed to have all capabilities without having to change their link flags As an example of linking with hypre a user may refer to the Makefile in the examples subdi rectory It is designed to build codes similar to user applications that link with and call hypre All include and linking flags are defined in the Makefile config file by configure 7 5 Error Flags Every hypre function returns an integer which is used to indicate errors during execution Note that the error flag returned by a given function reflects the errors from all previous calls
47. dge orientation Since the gradient operator is normalized i e independent the edge finite element must also be normalized in the discretization This solver is currently developed for perfectly conducting boundary condition Dirichlet Hence the rows and columns of the matrix that corresponding to the grid boundary must be set to the identity or zeroed off This can be achieved with the routines HY PRE_SStruct MaxwellPhysBdy and HYPRE SStructMaxwellEliminateRowsCols The former identifies the ranks of the rows that are located on the grid boundary and the latter adjusts the boundary rows and cols As usual the rhs of the linear system must also be zeroed off at the boundary rows This can be done using HYPRE_SStruct MaxwellZeroVector With the adjusted linear system and a gradient operator the user can form the Maxwell multigrid solver using several different edge interpolation schemes For problems with smooth coefficients the natural Nedelec interpolation operator can be used This is formed by calling HYPRE_SStructMaxwellSetConstantCoef with the flag gt 0 before setting up the solver otherwise the default edge interpolation is an operator collapsing element agglomeration scheme This is suit able for variable coefficients Also before setting up the solver the user must pass the gradient oper ator whether user or HY PRE_MaxwellGrad generated with HY PRE_SStructMaxwellSetGrad Af ter these preliminary calls the Maxwell solver can b
48. direction on part 3 corresponds to the z direction on part 2 The Assemble routine is a collective call i e must be called on all processes from a common synchronization point and finalizes the grid assembly making the grid ready to use With the neighbor information it is now possible to determine where off part stencil entries couple Take for example any shared part boundary such as the boundary between parts 2 and 3 Along these boundaries some stencil entries reach outside of the part If no neighbor information is given these entries are effectively zeroed out i e they don t participate in the discretization However with the additional neighbor information when a stencil entry reaches into a neighbor box it is then coupled to the part described by that neighbor box information Another important consequence of the use of the SetNeighborbox routine is that it can de clare variables on different parts as being the same For example the face variables on the boundary of parts 2 and are recognized as being shared by both parts prior to the SetNeighborbox call there were two distinct sets of variables Note also that these variables are of different types on the two parts on part 2 they are x face variables but on part they are y face variables For brevity we consider only the description of the y face stencil in Figure ie the third stencil in the figure To do this the stencil entries are assigned unique la
49. e composite matrix can be formed Since the AMR levels correspond to semi structured grid parts the composite matrix is a semi structured matrix consisting of structured components within each part and unstructured com ponents describing the coarse to fine fine to coarse connections The structured and unstructured components can be set using stencils and the HYPRE_SStructGraphAddEntries routine respec tively The matrix coefficients can be filled after setting these non zero patterns Between each pair of successive AMR levels the coarse matrix underlying the refinement patch must be the identity and the corresponding rows of the rhs must be zero These can performed using routines HYPRE SStructFACZeroCFSten to zero off the stencil values reaching from coarse boxes into refinement boxes HYPRE SStructFACZeroFCSten to zero off the stencil values reaching from refinement boxes into coarse boxes HYPRE SStructFACZeroAMRMatrixData to set the identity at coarse grid points underlying a refinement patch SStructFACZeroAMRVectorData to zero off a vector at coarse grid points underlying a refinement patch These routines can sim plify the user s matrix setup For example consider two successive AMR levels with the coarser level consisting of one box and the finer level consisting of a collection of boxes Rather than dis tinguishly setting the stencil values and the identity in the appropriate locations the user can set the stenci
50. e setup by calling HYPRE_SStruct MaxwellSetup There are two solver cycling schemes that can be used to solve the linear system To describe these one needs to consider the augmented system operator Aee Aen A 6 2 Ann where Aee is the stiffness matrix corresponding to the above curl curl formulation Ann is the nodal Poisson operator created by taking the Galerkin product of Aee and the gradient operator and Ane and Aen are the nodal edge coupling operators see 12 The algorithm for this Maxwell solver is based on forming a multigrid hierarchy to this augmented system using the block diagonal interpolation operator 0 P 0 Pr where are respectively the edge and nodal interpolation operators determined individ ually from Aee and Ann Taking a Galerkin product between A and P produces the next coarse augmented operator which also has the nodal edge coupling operators Applying this procedure re cursively produces nodal edge coupling operators at all levels Now the first solver cycling scheme 6 7 HYBRID Al HYPRE_SStructMaxwellSolve keeps these coupling operators on all levels of the V cycle The second cheaper scheme HY PRE_SStructMaxwellSolve2 keeps the coupling operators only on the finest level i e separate edge and nodal V cycles that couple only on the finest level 6 7 Hybrid The hybrid solver is designed to detect whether a multigrid preconditioner is needed when solving a linear
51. e to use AMS as preconditioner only After the above calls the solver is ready to be constructed The user has to provide the stiffness matrix A in ParCSR format and the hypre parallel vectors b and x The vectors are actually not used in the current AMS setup The setup call reads HYPRE_AMSSetup solver A b x It is important to note the order of the calling sequence For example do not call HYPRE_AMSSetup before calling HYPRE_AMSSetDiscreteGradient and one of the functions HYPRE_AMSSetCoordinateVectors or HYPRE_AMSSetEdgeConstantVectors Once the setup has completed we can solve the linear system by calling HYPRE_AMSSolve solver A b x Finally the solver can be destroyed with HYPRE_AMSDestroy amp solver More details can be found in the files ams h and ams c located in the parcsr 1s directory 6 10 The MLI Package MLI is an object oriented module that implements the class of algebraic multigrid algorithms based on Vanek and Brezina s smoothed aggregation method 22 21 There are two main algorithms in this module the original smoothed aggregation algorithm and the modified version that uses the finite element substructure matrices to construct the prolongation operators As such the later algorithm can only be used in the finite element context via the finite element interface In addition the nodal coordinates obtained via the finite element interface can be used to construct a better prolongation operator t
52. ective call with each process passing its own index extents jlower and jupper The names 34 CHAPTER 5 LINEAR ALGEBRAIC SYSTEM INTERFACE IJ of these extent parameters begin with a j because we typically think of matrix vector multiplies as the fundamental operation involving both matrices and vectors For matrix vector multiplies the vector partitioning should match the column partitioning of the matrix which also uses the j notation For linear system solves these extents will typically match the row partitioning of the matrix as well The SetObjectType routine sets the underlying vector storage type to HYPRE_PARCSR this is the only storage type currently supported The Initialize routine indicates that the vector coefficients or values are ready to be set This routine may or may not involve the allocation of memory for the coefficient data depending on the implementation The SetValues routine sets the vector values for some number nvalues of indices Each process should set only those vector values that it owns in the data distribution The Assemble routine is a trivial collective call and finalizes the vector assembly making the vector ready to use The GetObject routine retrieves the built vector object so that it can be passed on to hypre solvers that use the ParVector internal storage format Vector values can be modified in much the same way as with matrices by first re initializing the vector with the In
53. ed to increase with global problem size Experimental results have shown that PILU preconditioning is in general more effective than Block Jacobi preconditioning for minimizing total solution time For scaled problems the relative advantage appears to increase as the number of processors is scaled upwards Euclid may also be used to good advantage as a smoother within multigrid methods 6 12 1 Overview Euclid is best thought of as an extensible ILU preconditioning framework Extensible means that Euclid can and eventually will time and contributing agencies permitting support many variants of ILU k and ILUT preconditioning The current release includes Block Jacobi ILU k and Parallel ILU k methods Due to this extensibility and also because Euclid was developed independently of the hypre project the methods by which one passes runtime parameters to Euclid preconditioners differ in some respects from the hypre norm While users can directly set options within their code options can also be passed to Euclid preconditioners via command line switches and or small text based configuration files The latter strategies have the advantage that users will not need to alter their codes as Euclid s capabilities are extended The following fragment illustrates the minimum coding required to invoke Euclid precondition ing within hypre application contexts The next subsection provides examples of the various ways in which Euclid s options can
54. erences but this example shows how similar the interfaces can be The first example is for the C only interface and the other examples are for the Babel interface in the C C Fortran and Python languages HYPRE IJVectorlInitialize b bHYPRE IJParCSRVector Initialize b amp ex b Initialize 65 66 CHAPTER 8 BABEL BASED INTERFACES call bHYPRE_IJParCSRVector_Initialize_f b ierr ex b Initialize 8 3 Interfaces Are Different Function names usually differ in minor ways as decribed in the following section 8 4 But sometimes they differ more for example to apply a solver to solve linear equations you say Solve in the C only interface and Apply in the Babel based interface For information on particular functions see the reference manuals or example files Function argument lists look mainly the same The data types of many arguments are different as required for example in C a preconditioner would be a HYPRE_Solver in the C only interface or a bHYPRE_Solver in the Babel based interface Sometimes there are greater differences the greatest ones are discussed below Again see the reference manuals or example files for more specific information In C or Fortran the Babel based interface requires an extra argument at the end called the exception It is not necessary for C or Python You should ignore it but it has to be there In C you should declare it as follows include sidl_Exception h sidl BaseIn
55. es of calls for entering information about your problem into hypre 3 Build solvers and preconditioners and set solver parameters optional Some parameters like convergence tolerance are the same across solvers while others are solver specific 4 Call the solve function for the solver 5 Retrieve desired information from solver Depending on your application there may be different things you may want to do with the solution vector Also performance information such as number of iterations is typically available though it may differ from solver to solver 6 CHAPTER 1 INTRODUCTION The subsequent chapters of this User s Manual provide the details needed to more fully under stand the function of each conceptual interface and each solver Remember that a comprehensive list of all available functions is provided in the hypre Reference Manual and the provided example codes may prove helpful as templates for your specific application Chapter 2 Structured Grid System Interface Struct In order to get access to the most efficient and scalable solvers for scalar structured grid applications users should use the Struct interface described in this chapter This interface will also provide access this is not yet supported to solvers in hypre that were designed for unstructured grid applications and sparse linear systems in general These additional solvers are usually provided via the unstructured grid interface FEI or the linear al
56. f the grid The Create routine creates an empty 2D grid object with five parts that lives on the MPI_COMM_WORLD communicator The SetExtents routine adds a new box to the grid The SetVariables routine associates three variables of type cell centered x face and y face with part 3 At this stage the description of the data on part 3 is complete However the spatial relationship between this data and the data on neighboring parts is not yet defined To do this we need to relate the index space for part 3 with the index spaces of parts 2 and 4 More specifically we need to tell the interface that the two grey boxes neighboring part 3 in the bottom right of Figure 3 4 also correspond to boxes on parts 2 and 4 This is done through the two calls to the SetNeighborbox routine We will discuss only the first call which describes the grey box on the right of the figure Note that this grey box lives outside of the box extents for the grid on part 3 but it can still be described using the index space for part 3 recall Figure 2 2 That is the grey box has extents 1 0 and 4 0 on part 3 s index space which is outside of part 3 s grid The arguments for the SetNeighborbox call are simply the lower and upper indices on part 3 and the corresponding indices on part 2 The final argument to the routine indicates that the x direction on part i e the i component of the tuple i corresponds to the y direction on part 2 and that the y
57. gebraic interface IJ described in Chapters and Figure gives an example of the type of grid currently supported by the Struct interface The interface uses a finite difference or finite volume style and currently supports only scalar PDEs 1 one unknown per gridpoint There are four basic steps involved in setting up the linear system to be solved 1 set up the grid 2 set up the stencil 3 set up the matrix 4 set up the right hand side vector process 0 process 1 6 4 C31 Figure 2 1 An example 2D structured grid distributed accross two processors 8 CHAPTER 2 STRUCTURED GRID SYSTEM INTERFACE STRUCT d y Index Space Figure 2 2 A box is a collection of abstract cell centered indices described by its minimum and maximum indices Here two boxes are illustrated To describe each of these steps in more detail consider solving the 2D Laplacian problem P NE j V u f in the domain 2 1 0 on the boundary Assume 2 1 is discretized using standard 5 pt finite volumes on the uniform grid pictured in 2 1 and assume that the problem data is distributed across two processes as depicted 2 1 Setting Up the Struct Grid The grid is described via a global index space i e via integer singles in 1D tuples in 2D or triples in 3D see F igure 2 2 The integers may have any value negative or posi
58. h is short for Scientific Interface Definition Language For more information about the SIDL language visit the Babel project website For the briefest possible description of the overall ideas of the hypre object system see Figure 8 1 Babel translates this object structure as appropriate for the language the user uses from SIDL s interfaces and classes to C classes or to mangled names in C and Fortran For exam ple the StructVector class defined in Interfaces idl can appear as a C class also named StructVector in the bHYPRE namespace as a C struct named bHYPRE StructVector or as a Fortran integer 8 Its member function SetValues can appear in C as a member function SetValues in C as a function bHYPRE StructVector SetValues or in Fortran as a subroutine bHYPRE StructVector SetValues f The Babel interface goes beyond the hypre design in that a solver and matrix are both a kind of operator ie something which acts on a vector to compute another vector If a solver takes a preconditioner it is called a preconditioned solver There is no class or interface for preconditioners it is possible though not always wise to use any solver as a preconditioner In order for a solver to act as an operator to map a vector to a vector it must contain not only a solution method but also the equations to be solved So you supply the solver with a matrix while constructing the solver All solvers have very similar interfaces so most of
59. h that filter represents the fraction of the nonzero elements that are dropped For example if filter 0 9 then approximately 90 percent of the entries in the computed approximate inverse are dropped 6 11 2 Preconditioning Nearly Symmetric Matrices A nonsymmetric but definite and nearly symmetric matrix A may be preconditioned with a sym metric preconditioner M Using a symmetric preconditioner has a few advantages such as guar anteeing positive definiteness of the preconditioner as well as being less expensive to construct The nonsymmetric matrix A must be definite i e A AT 2 is SPD and the a priori sparsity pattern to be used must be symmetric The latter may be guaranteed by 1 constructing the sparsity pattern with a symmetric matrix or 2 if the matrix is structurally symmetric has symmetric pattern then thresholding to construct the pattern is not used i e zero value of the thresh parameter is used 6 12 Euclid The Euclid library is a scalable implementation of the Parallel ILU algorithm that was presented at SC99 10 and published in expanded form in the SIAM Journal on Scientific Computing 11 By scalable we mean that the factorization setup and application triangular solve timings remain 50 CHAPTER 6 SOLVERS AND PRECONDITIONERS nearly constant when the global problem size is scaled in proportion to the number of processors As with all ILU preconditioning methods the number of iterations is expect
60. h the Babel Interface You can build hypre almost the same way with and without the Babel based interface Normally the only difference is that the configure line needs an extra argument for example configure with babel make rather than configure make 74 CHAPTER 8 BABEL BASED INTERFACES The configure system will enable whatever built in languages it can find compilers for The configure system for the runtime portion of Babel included with hypre and enabled with the Babel based interface will automatically compile and run a few tiny test programs This has been a problem in multiprocessing AIX systems where compiled programs are normally run in a different environment from the configure system For AIX systems with POE the hypre distribution includes a workaround script nopoe When necessary build hypre as follows instead of the above nopoe configure with babel make 8 11 1 Building HYPRE with Python Using the Babel Interface To build the hypre library so you can call it from Python code the first step is to build a suitable Python Start with a recent version of Python with the Numeric Python extension for details see the section on recommended external software in the Babel Users Manual We have only tested hypre s Python interface with pyMPI a Python extension which supports MPI It is likely that you can make it work with other MPI extensions or even no MPI at all If you try this let us know how it works You
61. han the pure translation modes Below is an example on how to set up MLI as a preconditioner for conjugate gradient HYPRE_LSI_MLICreate MPI_COMM_WORLD amp pcg_precond HYPRE_LSI_MLISetParams pcg_precond MLI strengthThreshold 0 08 HYPRE_PCGSetPrecond pcg_solver HYPRE_PtrToSolverFcn HYPRE_LSI_MLISolve HYPRE_PtrToSolverFcn HYPRE_LSI_MLISetup pcg_precond 48 CHAPTER 6 SOLVERS AND PRECONDITIONERS Note that parameters are set via HYPRE LSI MLISetParams A list of valid parameters that can be set using this routine can be found in the FEI section of the reference manual 6 11 ParaSails ParaSails is a parallel implementation of a sparse approximate inverse preconditioner using a priori sparsity patterns and least squares Frobenius norm minimization Symmetric positive definite SPD problems are handled using a factored SPD sparse approximate inverse General nonsymmetric and or indefinite problems are handled with an unfactored sparse approximate inverse It is also possible to precondition nonsymmetric but definite matrices with a factored SPD preconditioner ParaSails uses a priori sparsity patterns that are patterns of powers of sparsified matrices ParaSails also uses a post filtering technique to reduce the cost of applying the preconditioner In advanced usage not described here the pattern of the preconditioner can also be reused to generate preconditioners for different matrices in a sequence of linear solves
62. he largest LFIL elements in each row that remain Smaller values of tol lead to more accurate preconditioners but can also lead to increases in the time to calculate the preconditioner 6 14 FEI Solvers After the FEI has been used to assemble the global linear system as described in Chapter 4 a number of hypre solvers can be called to perform the solution This is straightforward if hypre s FEI has been used If an external FEI is employed the user needs to link with hypre s implementation of the LinearSystemCore class as described in Section 7 7 Solver parameters are specified as an array of strings and a complete list of the available options can be found in the FEI section of the reference manual They are passed to the FEI as in the following example nParams 5 paramStrings new char nParams for i 0 i lt nParams i paramStrings i new char 100 strcpy paramStrings 0 solver cg strcpy paramStrings 1 preconditioner diag strcpy paramStrings 2 maxiterations 100 strcpy paramStrings 3 tolerance 1 0e 6 strcpy paramStrings 4 outputLevel 1 feiPtr gt parameters nParams paramStrings To solve the linear system of equations we call feiPtr gt solve amp status where the returned value status indicates whether the solve was successful Finally the solution can be retrieved by the following function call feiPtr gt getBlockNodeSolution elemBlkID nNodes nodeIDList
63. he stencil numbers for each of the stencil entries None of the calls are collective calls 2 3 Setting Up the Struct Matrix The matrix is set up in terms of the grid and stencil objects described in Sections and The coefficients associated with each stencil entry will typically vary from gridpoint to gridpoint but in the example problem being considered they are as follows over the entire grid except at boundaries see below 1 2 2 On process 0 the code in Figure 2 6 will set up matrix values associated with the center entry 0 and south entry 3 stencil entries as given by 2 2 and Figure 2 6 boundaries are ignored here temporarily The Create routine creates an empty matrix object The Initialize routine indicates that the matrix coefficients or values are ready to be set This routine may or may not involve the allocation of memory for the coefficient data depending on the implementation The optional Set routines mentioned later in this chapter and in the Reference Manual should be called before this step The SetBoxValues routine sets the matrix coefficients for some set of stencil entries over the gridpoints in some box Note that the box need not correspond to any of the boxes used to create the grid but values should be set for all gridpoints that this process 2 3 SETTING UP THE STRUCT MATRIX 11
64. id based interfaces are not appropriate It requires more work on the user s part though still less than building par allel sparse data structures General solvers and preconditioners are available through this interface but not specialized solvers which need more information Our experience is that users with legacy codes in which they already have code for building matrices in particular formats find the IJ interface relatively easy to use See Chapter 5 for details Generally a user should choose the most specific interface that matches their application be cause this will allow them to use specialized and more efficient solvers and preconditioners without losing access to more general solvers For example the second row of Figure 1 is a set of linear solver algorithms Each linear solver group requires different information from the user through the conceptual interfaces So the geometric multigrid algorithm GMG listed in the left most box 1 3 HOW TO GET STARTED 5 for example can only be used with the left most conceptual interface On the other hand the ILU algorithm in the right most box may be used with any conceptual interface Matrix requirements for each solver and preconditioner are provided in Chapter and in the hypre Reference Manual Your desired solver strategy may influence your choice of conceptual interface A typical user will select a single Krylov method and a single preconditioner to solve their system
65. ified the default directories src hypre lib and src hypre include are used 7 2 1 Configure Options There are many options to configure to allow the user to override and refine the defaults for any system The best way to find out what options are available is to display the help package by executing configure help which also includes the usage information The user can mix and match the configure options and variable settings to meet their needs Some of the commonly used options include enable debug Sets compiler flags to generate information needed for debugging enable shared Build shared libraries NOTE in order to use the resulting shared libraries the user MUST have the path to the libraries defined in the environment variable LD LIBRARY PATH with print errors Print HYPRE errors with no global partitioning Do NOT store a global partition of the data NOTE this option reduces storage and improves performance when using many thousands of processors not recommended for 1000 processors The user can mix and match the configure options and variable settings to meet their needs It should be noted that hypre can be configured with external BLAS and LAPACK libraries which can be combined with any other option This is done as follows currently both libraries must be configured as external together configure with blas lib blas lib name with blas lib dirs path to blas lib with lapack lib lapack
66. in the composite grid matrix is typically the composition of an interpolation rule and a discretization formula In this example we use a simple piecewise constant interpolation i e the solution value in a coarse cell is equal to the solution value at the cell center Then the flux across a portion of the coarse fine interface is approximated by a difference of the solution values on each side As an example consider approximating the flux across the left interface of cell 6 6 in Figure Let h be the coarse grid mesh size and consider a local coordinate system with the origin at the center of cell 6 6 We approximate the flux as follows h u 0 0 u 3h 4 0 2 3h 4 Q WIN N 4 ug h A4 s ds h A uz h 4 0 3 2 us u2 3 3 2 STRUCTURED ADAPTIVE MESH REFINEMENT 25 2 3 6 6 Figure 3 8 Coupling for equation at corner of refinement patch Black lines solid and broken are stencil couplings Gray line are non stencil couplings The first approximation uses the midpoint rule for the edge integral the second uses a finite difference formula for the derivative and the third the piecewise constant interpolation to the solution in the coarse cell This means that the equation for the variable at cell 6 6 involves not only the stencil couplings to 6 7 and 7 6 on part 1 but also non stencil couplings to 2 3 and 3 2 on part 0 These non stencil couplings are de
67. include formation of non Galerkin coarse grid operators i e coarse grid operators underlying refinement patches are automatically generated and non stored linear constant inter polation restriction operators Implementation particularities include a processor redistribution of the generated coarse grid operators so that intra level communication between adaptive mesh refinement AMR levels during the solve phase is kept to a minimum This redistribution is hidden from the user 6 6 MAXWELL 39 The user input is essentially a linear system describing the composite operator and the refine ment factors between the AMR levels To form this composite linear system the AMR grid is described using semi structured grid parts Each AMR level grid corresponds to a separate part so that this level grid is simply a collection of boxes all with the same refinement factor i e it is a struct grid However several restrictions are imposed on the patch box refinements First a refinement box must cover all underlying coarse cells i e refinement of a partial coarse cell is not permitted Also the refined coarse indices must follow a mapping with ri ro r3 denoting the refinement factor and a1 a2 a3 x b1 b2 b3 denoting the coarse subbox to be refined the mapping to the refined patch is r1 a1 T2 a3 x ri b try 1 T2 b2 r2 1 73 x b3 1 With the AMR grid constructed under these restrictions th
68. ing is likely to cause the filled matrix F L U I to be unsymmetric This setting has no effect when ILUT factorization is selected rowScale Scale values prior to factorization such that the largest value in any row is 1 or 1 Default 0 false CAUTION If the coefficient matrix A is symmetric this setting is likely to cause the filled matrix F D U I to be unsymmetric Guidance if the matrix is poorly scaled turning on row scaling may help convergence ilut float Use ILUT factorization instead of the default ILU k Here float is the drop tolerance which is relative to the largest absolute value of any entry in the row being factored CAUTION If the coefficient matrix A is symmetric this setting is likely to cause the filled matrix F 2 0 I to be unsymmetric maxNzPerRow int This sets the maximum number of nonzeros that is permitted in any row of F in addition to the number that would result from an ILU 0 factorization A negative value indicates infinity no limit This setting is effective for both ILU k and ILUT factorization methods Default infinity for ILU k 5 for ILUT 6 13 PILUT Parallel Incomplete Factorization Note this code is no longer supported by the hypre team We recommend to use Euclid instead which is more versatile and in general more efficient especially when used with many processors PILUT is a parallel preconditioner based on Saad s dual threshold incomplete fac
69. itialize routine 5 3 A Scalable Interface As explained in the previous sections problem data is passed to the hypre library in its distributed form However as is typically the case for a parallel software library some information regarding the global distribution of the data will be needed for hypre to perform its function In particular a solver algorithm requires that a processor obtain nearby data from other processors in order to complete the solve While a processor may easily determine what data it needs from other processors it may not know which processor owns the data it needs Therefore processors must determine their communication partners or neighbors The straightforward approach to determining neighbors involves constructing a global partition of the data which requires O P data storage This storage requirement as well the costs of many of the associated algorithms that access the storage is not scalable for machines such as BlueGene L with tens of thousands of processors The problem of determining inter processor communication in the absence of a global description of the data in a scalable manner is addressed in 2 When using hypre on many thousands of processors compiling the library with the no global partition option as detailed in Section 7 2 1 improves scalability as shown in 2 Note that this optimization is only recommended when using at least several thousand of processors and is most beneficial
70. ization scheme for quadrilateral r z meshes J Comp Physics 144 17 51 1998 J W Ruge and K St ben Algebraic multigrid AMG In S F McCormick editor Multigrid Methods volume 3 of Frontiers in Applied Mathematics pages 73 130 SIAM Philadelphia PA 1987 5 Schaffer A semi coarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients SIAM J Sci Comput 20 1 228 242 1998 K St ben Algebraic multigrid AMG an introduction with applications In U Trottenberg C Oosterlee and A Sch ller editors Multigrid Academic Press 2001 P Vanek M Brezina and J Mandel Convergence of algebraic multigrid based on smoothed aggregation Numerische Mathematik 88 559 579 2001 P Van k J Mandel and M Brezina Algebraic multigrid based on smoothed aggregation for second and fourth order problems Computing 56 179 196 1996 U M Yang On the use of relaxation parameters in hybrid smoothers Numerical Linear Algebra with Applications 11 155 172 2004 BIBLIOGRAPHY TT 24 U M Yang Parallel algebraic multigrid methods high performance preconditioners In A M Bruaset and A Tveito editors Numerical Solution of Partial Differential Equations on Parallel Computers pages 209 236 Springer Verlag 2006 Also available as LLNL technical report UCRL BOOK 208032
71. l No need to add non stencil entries in this example HYPRE SStructGraphAssemble graph Figure 3 6 Test figure 24 CHAPTER 3 SEMI STRUCTURED GRID SYSTEM INTERFACE SSTRUCT e e e 7 9 9 9 e e e e e e e 86 part 1 part 0 Figure 3 7 Structured AMR grid example Shaded regions correspond to process 0 unshaded to process 1 The grey dots are dummy variables In the example parts are distributed across the same two processes with process 0 having the left half of both parts The composite grid is then set up part by part by making calls to GridSetExtents just as was done in Section 3 1 and Figure 3 4 no SetNeighborBox calls are made in this example Note that in the interface there is no required rule relating the indexing on the refinement patch to that on the global coarse grid they are separate parts and thus each has its own index space In this example we have chosen the indexing such that refinement cell 27 27 lies in the lower left quadrant of coarse cell i j Then the stencil is set up In this example we are using a finite volume approach resulting in the standard 5 point stencil in Figure in both parts The grid and stencil are used to define all intra part coupling in the graph the non zero pattern of the composite grid matrix The inter part coupling at the coarse fine interface is described by GraphAddEntries calls This coupling
72. l edge space and subspace corrections based on Bg and By We can employ a number of options here utilizing various combinations of the smoother and solvers in additive or multiplicative fashion If 8 is identically zero one can skip the subspace correction associated with G in which case the solver is a two level method 6 9 2 Sample Usage AMS can be used either as a solver or as a preconditioner Below we list the sequence of hypre calls needed to create and use it as a solver We start with the allocation of the HYPRE_Solver object HYPRE_Solver solver HYPRE AMSCreate amp solver Next we set a number of solver parameters Some of them are optional while others are necessary in order to perform the solver setup AMS offers the option to set the space dimension By default we consider the dimension to be 3 The only other option is 2 and it can be set with the function given below We note that a 3D solver will still work for a 2D problem but it will be slower and will require more memory than necessary 6 9 AMS 45 HYPRE_AMSSetDimension solver dim The user is required to provide the discrete gradient matrix G AMS expects a matrix defined on the whole mesh with no boundary edges nodes excluded It is essential to not impose any boundary conditions on G Regardless of which hypre conceptual interface was used to construct G one can obtain a ParCSR version of it This is the expected format in the following function HYPRE_AM
73. l requirements such as access to the matrix coefficients 2 Create the solver preconditioner via the Create routine 3 Choose parameters for the preconditioner and or solver Parameters are chosen through the Set calls provided by the solver Throughout hypre we have made our best effort to give all parameters reasonable defaults if not chosen However for some precondi tioners solvers the best choices for parameters depend on the problem to be solved We give recommendations in the individual sections on how to choose these parameters Note that in hypre convergence criteria can be chosen after the preconditioner solver has been setup For a complete set of all available parameters see the Reference Manual 35 36 CHAPTER 6 SOLVERS AND PRECONDITIONERS System Interfaces Solvers Struct SStruct FEI IJ Jacobi X X SMG X X PFMG X X SysPFMG X Split X FAC X Maxwell X BoomerAMG X X X AMS X X IX MLI X X IX ParaSails X X X Euclid X X X PILUT X X IX PCG X X X X GMRES X X X X BiCGSTAB X X X X Hybrid X X X X Table 6 1 Current solver availability via hypre conceptual interfaces 4 Pass the preconditioner to the solver For solvers that are not preconditioned this step is omitted The preconditioner is passed through the SetPrecond call 5 Set up the solver This is just the Setup routine At this point the matrix and right hand side is passed into the solver or preconditioner N
74. l values on the whole coarse grid using the HYPRE SStructMatrixSetBox Values routine and then zero off the appropriate values using the above zeroing routines The coarse matrix underlying these patches are algebraically generated by operator collapsing the refinement patch operator and the fine to coarse coefficients this is why stencil values reaching out of a part must be zeroed This matrix is re distributed so that each processor has all of its coarse grid operator To solve the coarsest AMR level a PFMG V cycle is used Note that a minimum of two AMR levels are needed for this solver 6 6 Maxwell Maxwell is a parallel solver for edge finite element discretization of the curl curl formulation of the Maxwell equation gt 0 40 CHAPTER 6 SOLVERS AND PRECONDITIONERS on semi structured grids Details of the algorithm can be found in 12 The solver can be viewed as an operator dependent multiple coarsening algorithm for the Helmholtz decomposition of the error correction Input to this solver consist of only the linear system and a gradient operator In fact if the orientation of the edge elements conforms to a lexigraphical ordering of the nodes of the grid then the gradient operator can be generated with the routine HY PRE_MaxwellGrad at grid points i j k and i 1 j k the produced gradient operator takes values 1 and 1 respectively which is the correct gradient operator for the appropriate e
75. les types cell centered z face and y face Discretization stencils for the cell centered left z face middle and y face right variables are also pictured Figure 3 3 Test figure 20 CHAPTER 3 SEMI STRUCTURED GRID SYSTEM INTERFACE SSTRUCT Oe A e A 94 94 HYPRE_SStructGrid grid int ndim 2 nparts 5 nvars 3 part 3 int extents 2 1 1 4 4 int vartypes HYPRE SSTRUCT VARIABLE CELL HYPRE SSTRUCT VARIABLE XFACE HYPRE SSTRUCT VARIABLE YFACE int nb2 n part 2 nb4_n_part 4 int nb2_exts 2 1 0 4 03 nb4_exts 2 0 1 0 4 int nb2_n_exts 2 441 1 1 4 nb4_n_exts 2 4 1 4 4 int nb2 map 2 1 0 nb4_map 2 0 1 HYPRE SStructGridCreate MPI COMM WORLD ndim nparts amp grid Set grid extents and grid variables for part 3 HYPRE SStructGridSetExtents grid part extents 0 extents 1 HYPRE SStructGridSetVariables grid part nvars vartypes Set spatial relationship between parts 3 and 2 then parts 3 and 4 HYPRE SStructGridSetNeighborBox grid part nb2 exts 0 nb2 exts 1 nb2 n part nb2 n exts 0 nb2 n exts 1 nb2 map HYPRE SStructGridSetNeighborBox grid part nb4_exts 0 nb4 exts 1 nb4 n part nb4 n exts 0 nb4 n exts 1 nb4 map HYPRE_SStructGridAssemble grid Figure 3 4 Code on process 3 for setting up the grid in Figure 8 2 3 1 BLOCK STRUCTURED GRIDS 21 with the rest o
76. linear systems of equations on massively parallel computers The hypre library was created with the primary goal of providing users with advanced parallel preconditioners The library features parallel multigrid solvers for both structured and unstructured grid problems For ease of use these solvers are accessed from the application code via hypre s conceptual linear system interfaces abbreviated to conceptual interfaces throughout much of this manual which allow a variety of natural problem descriptions This introductory chapter provides an overview of the various features in hypre discusses further sources of information on hypre and offers suggestions on how to get started 1 1 Overview of Features e Scalable preconditioners provide efficient solution on today s and tomorrow s sys tems hypre contains several families of preconditioner algorithms focused on the scalable solution of very large sparse linear systems Note that small linear systems systems that are solvable on a sequential computer and dense systems are all better addressed by other libraries that are designed specifically for them hypre includes grey box algorithms that use more than just the matrix to solve certain classes of problems more efficiently than general purpose libraries This includes algorithms such as structured multigrid e Suite of common iterative methods provides options for a spectrum of problems hypre provides several of the most commonl
77. lize routine indicates that the vector co efficients or values are ready to be set This routine follows the same rules as its corresponding Matrix routine The SetBoxValues routine sets the vector coefficients over the gridpoints in some box and again follows the same rules as its corresponding Matrix routine The Assemble 14 CHAPTER 2 STRUCTURED GRID SYSTEM INTERFACE STRUCT HYPRE StructVector b double values 18 int i HYPRE_StructVectorCreate MPI_COMM_WORLD grid amp b HYPRE StructVectorInitialize b for i 0 i lt 18 i 1 values i 0 0 HYPRE StructVectorSetBoxValues b ilower 0 iupper 0 values HYPRE StructVectorSetBoxValues b ilower 1 iupper 1 values HYPRE StructVectorAssemble b Figure 2 8 Code for setting up right hand side vector values routine is a collective call and finalizes the vector assembly making the vector ready to use 2 5 Symmetric Matrices Some solvers and matrix storage schemes provide capabilities for significantly reducing memory usage when the coefficient matrix is symmetric In this situation each off diagonal coefficient appears twice in the matrix but only one copy needs to be stored The Struct interface provides support for matrix and solver implementations that use symmetric storage via the SetSymmetric routine To describe this in more detail consider again the 5 pt finite volume discretization of on the grid pictured in Figure 2 1 Because
78. look like bHYPRE Class Function f In the C Babel interface SIDL Babel classes are actually implemented as classes so functions generally look like Instance Function where Instance is an instance of the class In C the Babel interface functions live in the namespace ucxx bHYPRE This function naming pattern is slightly broken in the case of service functions provided by Babel rather than hypre All of them contain a double underscore __ These functions are used for casting certain arrays and sometimes memory management For more information see sections and ET Most functions are member functions of a class In non object oriented languages the first argument of the function will be the object which owns the function For example to solve equations with a PCG solver you apply it to vectors using its member function Apply In C Fortran and Python this concept is expressed as pcg_solver Apply b x C in the ucxx bHYPRE namespace bHYPRE PCG Apply PCG solver b amp x ex C call bHYPRE PCG Apply f PCG solver b x ierr ex Fortran peg solver Apply b x Python In the Babel interface data types usually look like Class in C bHYPRE Class in C and integer 8 in Fortran Similar data types in the C only interface would look like HYPRE Class Here Class is a class name where the interface is defined in the SIDL file Interfaces idl Sometimes the class name is slightly different in the
79. mmunicator The value of symmetry has the following meanings to indicate the symmetry and definiteness of the problem and to specify the type of preconditioner to construct 6 12 EUCLID 49 value meaning 0 nonsymmetric and or indefinite problem and nonsymmetric preconditioner 1 SPD problem and SPD factored preconditioner 2 nonsymmetric definite problem and SPD factored preconditioner For more information about the final case see section 6 11 2 Parameters for setting up the preconditioner are specified using int HYPRE_ParaSailsSetParams HYPRE_Solver solver double thresh int nlevel double filter The parameters are used to specify the sparsity pattern and filtering value see above and are described with suggested values as follows parameter type range sug values default meaning nlevel integer nlevel gt 0 0 1 2 1 m nlevel 1 thresh real thresh gt 0 0 0 1 0 01 0 1 thresh thresh thresh lt 0 0 75 0 90 thresh selected automatically filter real filter gt 0 0 0 05 0 001 0 05 filter value filter filter lt 0 0 90 filter value selected automatically When thresh lt 0 then a threshold is selected such that thresh represents the fraction of the nonzero elements that are dropped For example if thresh 0 9 then A will contain approximately ten percent of the nonzeros in A When filter lt 0 then a filter value is selected suc
80. o be obtained directly between calls to other hypre functions by calling HYPRE GetError If an argument error ERROR ARG has occurred the argument index counting from 1 can be obtained from GetErrorArgO get a character string with a description of all errors in a given error flag use HYPRE DescribeError int hypre ierr char descr Finally if hypre was configured with with print errors additional error information will be printed to the standard error during execution 7 6 Bug Reporting and General Support Simply send and email to hypre support 11n1 gov to report bugs request features or ask general usage questions An issue number will be assigned to your email so that we can track progress we are using an issue tracking tool called Roundup to do this Users should include as much relevant information as possible in their issue emails including at a minimum the hypre version number being used For compile and runtime problems please also include the machine type operating system MPI implementation compiler and any error messages produced 7 7 Using in External FEI Implementations To set up hypre for use in external e g Sandia s FEI implementations one needs to follow the following steps 1 obtain the hypre and Sandia s FEI source codes 2 compile Sandia s FEI fei 2 5 0 to create the fei_base library 3 compile hypre a unpack the archive and go into the src di
81. om your code which you can write in any of several languages The Struct Structured grid SStruct Semi Structured grid and IJ linear algebra style interfaces all can be used this way The code you write to call hypre functions through the Babel based interface works much the same as the code you write to call them through the original C interface The function names are different the object structure of hypre is more visible and there are many more minor differences This chapter will discuss these differences and present brief examples You can also see them in action by looking at the complete examples in the examples directory You do not need to have the Babel software to use the Babel based interface of hypre Prebuilt interfaces for the C C Fortran and Python languages are included with the hypre distribution We will enlarge the distribution with more such interfaces whenever hypre users indicate that they need them and our tools such as Babel support them Although you do not need to know about Babel to use the Babel based interface of hypre if you are curious about Babel you can look at its documentation at http www 1l1nl gov CASC components 8 2 Interfaces Are Similar In most respects the older C only interface and the Babel based interface are very similar The function names always differ a little and often there are minor differences in the argument lists The following sections will discuss the more significant diff
82. or users who are interested in a more detailed description and analysis of the solvers A complete list of all routines that are available can be found in the reference manual 6 1 SMG SMG is a parallel semicoarsening multigrid solver for the linear systems arising from finite difference finite volume or finite element discretizations of the diffusion equation V DVu ou f 6 1 on logically rectangular grids The code solves both 2D and 3D problems with discretization stencils of up to 9 point in 2D and up to 27 point in 3D See 7 for details on the algorithm and its parallel implementation performance SMG is a particularly robust method The algorithm semicoarsens in the z direction and uses plane smoothing The xy plane solves are effected by one V cycle of the 2D SMG algorithm which semicoarsens in the y direction and uses line smoothing 38 CHAPTER 6 SOLVERS AND PRECONDITIONERS 6 2 PFMG is a parallel semicoarsening multigrid solver similar to SMG See 7 for details on the algorithm and its parallel implementation performance The main difference between the two methods is in the smoother PFMG uses simple pointwise smoothing As a result PFMG is not as robust as SMG but is much more efficient per V cycle 6 3 SysPFMG SysPFMG is a parallel semicoarsening multigrid solver for systems of elliptic PDEs It is a gener alization of PFMG with the interpolation defined only within the same variable The rela
83. ote that the actual right hand side is not used until the actual solve is performed At this point the solver preconditioner is fully constructed and ready for use Use 1 Set convergence criteria Convergence can be controlled by the number of iterations as well as various tolerances such as relative residual preconditioned residual etc Like all parameters reasonable defaults are used Users are free to change these though care must be taken For example if an iterative method is used as a preconditioner for a Krylov method a constant number of iterations is usually required 2 Solve the system This is just the Solve routine 6 1 SMG 37 Finalize 1 Free the solver or preconditioner This is done using the Destroy routine Synopsis In general a solver let s call it SOLVER is set up and run using the following routines where A is the matrix b the right hand side and x the solution vector of the linear system to be solved Create Solver int HYPRE_SOLVERCreate MPI_COMM_WORLD amp solver set certain parameters if desired HYPRE_SOLVERSetTol solver 1 e 8 Set up Solver HYPRE_SOLVERSetup solver A b x Solve the system HYPRE_SOLVERSolve solver A b x Destroy the solver HYPRE_SOLVERDestroy solver In the following sections we will give brief descriptions of the available hypre solvers with some suggestions on how to choose the parameters as well as references f
84. oth ing 1 stands for BoomerAMG based on and 2 refers to a call to BoomerAMG for The abbreviation zyyz for x y z 0 1 2 refers to a multiplicative subspace correction based solvers 2 y y and z in that order The abbreviation x y z stands for an additive subspace correction method based on y and z solvers The additive cycles are meant to be used only when AMS is called as a preconditioner In our experience the choices cycle type 1 and cycle type 5 often produced fastest solution times while cycle type 7 resulted in smallest number of iterations Additional solver parameters as the maximum number of iterations the convergence tolerance and the output level can be set with HYPRE AMSSetMaxIter solver maxit default value 20 HYPRE AMSSetTol solver tol default value 1e 6 HYPRE AMSSetPrintLevel solver print default value 1 More advanced parameters affecting the smoothing and the internal AMG solvers can be set with the following three functions HYPRE AMSSetSmoothing ptions solver 2 1 1 0 1 0 HYPRE AMSSetAlphaAMGOptions solver 10 1 3 0 25 HYPRE AMSSetBetaAMGOptions solver 10 1 3 0 25 For singular problems where 3 0 in the whole domain different in fact simpler version of the AMS solver is offered To allow for this simplification use the following hypre call HYPRE_AMSSetBetaPoissonMatrix solver NULL If 0 is zero only in parts of the domain the p
85. ox ml RUE RE ORE Rom Geo d AGE CADRE AERE REV ERR ake yor 62 4 4a Sm RS Ea BER EOE OG EE OE REO 6 8 BoomerAMG 6 8 1 Parameter 8 6 9 604 64 4b i a TT 43 6 9 1 OvervieW 44 4 2 84 wk E E Ge RUE ek ee BOE won e E eos 6 9 2 Sample 44 6 10 The MLI 47 Gol sx utum od te Ee we n ee ee ee 6111 Parameter Settings o dee so esonero 2x 9 og NUR Tout te od OD OR 6 11 2 Preconditioning Nearly Symmetric 49 612 EET 49 6 12 1 Overview a sas cu saaa moa au 9 3o dox 9 Xd 6 12 2 Setting Options Examples 6 12 3 Options Summary 2 2 2 2 2 2422 6 13 PILUT Parallel Incomplete Factorization 6 14 FEISolver o u EO Y Rog Eo 9 OR n 6 14 1 Solvers Available Only through General Information 7 1 Getting the Source 7 2 Building the Library gt sspe dem KE Eos ded tereo Oe E RULES Rede 7 2 1 Configure Options foo Make Targ tS Ln oa hte bo eh
86. per the following example 52 CHAPTER 6 SOLVERS AND PRECONDITIONERS mpirun np 2 phoo level Since Euclid looks for the database file when HYPRE_EuclidCreate is called and parses the command line when HYPRE_EuclidSetParams is called option values passed on the command line will override any similar settings that may be contained in the database file Also if same option name appears more than once on the command line the final appearance determines the setting Some options such as bj see next subsection are boolean Euclid always treats these options as the value 1 true or 0 false When passing boolean options from the command line the value may be committed in which case it assumed to be 1 Note however that when boolean options are contained in a configuration file either the 1 or 0 must stated explicitly Method 3 There are two ways in which you can read in options from a file whose name is other than database First you can call HYPRE EuclidSetParamsFromFile to specify a configuration filename Second if you have passed the command line arguments as described above in Method 2 you can then specify the configuration filename on the command line using the db filename filename option e g mpirun np 2 phoo db filename myConfigFile 6 12 3 Options Summary level int Factorization level for ILU k Default 1 Guidance for 2D convection diffusion and simil
87. re aoaaa ee 1 3 2 Choosing a conceptualinterface a 1 3 8 Writing your code 2 Structured Grid System Interface Struct 2 1 Setting Up the Struct Grid 2 2 Setting Up the Struct Stencil 2 3 Setting Up the Struct Matrix 2 4 Setting Up the Struct Right Hand Side Vector 2 5 Symmetric Matrices 3 Semi Structured Grid System Interface SStruct 3 1 Block Structured Grids 3 2 Structured Adaptive Mesh Refinement 4 Finite Element Interface AV Tntroduction s aq qed BAR A GCaex Rep eee 35 4 2 A Brief Description of the Finite Element Interface 5 Linear Algebraic System Interface IJ 5 1 IJ Matrix Interface eee a 5 2 IJ Vector Interface eee ees 5 3 A Scalable Interface e ee 6 Solvers and Preconditioners SM um xx ese 02 eee ox ee d Rex Wm Eww Eom Ek x 6 3 SYSPPMG eals Be ew qe Rho RUE POR Or n X ON e RO Ros 6 4 SplitSolve sa saap aa ukaa a lll oh yo CONTENTS 65 soys hoe le Ge Get d A uo ee Hee ee RU d 38 6 0 Maxwell
88. rectory b do a configure with the with fei inc dir option set to the FEI include directory plus other compile options c compile with make install to create the HYPRE_LSI library in hypre 1lib 4 call the FEI functions in your application code as shown in Chapters 4 and 6 62 CHAPTER 7 GENERAL INFORMATION a include cfei hypre h in your file b include FEI_Implementation h in your file 5 Modify your Makefile a include hypre s include and 1ib directories in the search paths b Link with 1fei base 1HYPRE LSI Note that the order in which the libraries are listed may be important Building an application executable often requires linking with many different software packages and many software packages use some LAPACK and or BLAS functions In order to alleviate the problem of multiply defined functions at link time it is recommended that all software libraries are stripped of all LAPACK and BLAS function definitions These LAPACK and BLAS functions should then be resolved at link time by linking with the system LAPACK and BLAS libraries e g dxml on DEC cluster Both hypre and SuperLU were built with this in mind However some other software library files needed may have the BLAS functions defined in them To avoid the problem of multiply defined functions it is recommended that the offending library files be stripped of the BLAS functions 7 8 Calling HYPRE from Other Languages The hypre library can
89. roblem is still singular but the AMS solver will try to detect this and construct a non singular preconditioner Two additional matrices are constructed in the setup of the AMS method one corresponding to the coefficient and another corresponding to This may lead to prohibitively high memory requirements and the next two function calls may help to save some memory For example if the Poisson matrix with coefficient 3 denoted by Abeta is available then one can avoid one matrix construction by calling HYPRE_AMSSetBetaPoissonMatrix solver Abeta Similarly if the Poisson matrix with coefficient is available denoted by Aalpha the second matrix construction can also be avoided by calling AMSSetAlphaPoissonMatrix solver Aalpha Note the following regarding these functions e Both of them change their input More specifically the diagonal entries of the input matrix corresponding to eliminated degrees of freedom due to essential boundary conditions are penalized 6 10 THE MLI PACKAGE 47 e HYPRE_AMSSetAlphaPoissonMatrix forces the AMS method to use simpler but weaker in terms of convergence method With this option the multiplicative AMS cycle is not guaranteed to converge with the default parameters The reason for this is the fact the solver is not variationally obtained from the original matrix it utilizes the auxiliary Poisson like matrices Abeta and Aalpha Therefore it is recommended in this cas
90. s of these routines Likewise setting up the right hand side is similar to what is described in Section See the hypre reference manual for details An alternative approach for describing the above problem through the interface is to use the GraphAddEntries routine instead of the GridSetNeighborBox routine In this approach the five parts would be explicitly sewn together by adding non stencil couplings to the matrix graph The main downside to this approach for block structured grid problems is that variables along block boundaries are no longer considered to be the same variables on the corresponding parts that share these boundaries For example any face variable along the boundary between parts 2 and 3 in Figure 3 2 would represent two different variables that live on different parts To sew the parts together correctly we would need to explicitly select one of these variables as the representative that participates in the discretization and make the other variable a dummy variable that is decoupled from the discretization by zeroing out appropriate entries in the matrix All of these complications are avoided by using the GridSetNeighborBox for this example 3 2 Structured Adaptive Mesh Refinement We now briefly discuss how to use the SStruct interface in a structured AMR application Consider Poisson s equation on the simple cell centered example grid illustrated in FigureB 7 For structured AMR applications each refinemen
91. scribed by GraphAddEntries calls The syntax for this call is simply the part and index for both the variable whose equation is being defined and the variable to which it couples After these calls the non zero pattern of the matrix and the graph is complete Note that the west and south stencil couplings simply drop off the part and are effectively zeroed out currently this is only supported for the HYPRE_PARCSR object type and these values must be manually zeroed out for other object types see MatrixSetObjectType in the reference manual The remaining step is to define the actual numerical values for the composite grid matrix This can be done by either MatrixSetValues calls to set entries in a single equation or by MatrixSetBoxValues calls to set entries for a box of equations in a single call The syntax for the MatrixSetValues call is a part and index for the variable whose equation is being set and an array of entry numbers identifying which entries in that equation are being set The entry numbers may correspond to stencil entries or non stencil entries 26 CHAPTER 3 SEMI STRUCTURED GRID SYSTEM INTERFACE SSTRUCT Chapter 4 Finite Element Interface 4 1 Introduction Many application codes use unstructured finite element meshes This section describes an interface for finite element problems called the FEI which is supported in hypre Figure 4 1 Example of an unstructured mesh
92. set the up cycle down cycle or the coarsest grid with HYPRE_BoomerAMGSetCycleRelaxType To use the more complicated smoothers e g block Schwarz ILU smoothers it is necessary to use HYPRE_BoomerAMGSetSmoothType and HYPRE_BoomerAMGSetSmoothNumLevels There are further parameter choices for the individual smoothers which are described in the reference manual The default relaxation type is hybrid Gauss Seidel with CF relaxation relax first the C then the F points on the down cycle and FC relaxation on the up cycle Note that if BoomerAMG is used as a preconditioner for comjugate gradient it is necessary to use a symmetric smoother such as weighted Jacobi or hybrid symmetric Gauss Seidel If the users wants to solve systems of PDEs and can provide information on which variables belong to which function BoomerAMG s systems AMG version can also be used Functions that enable the user to access the systems AMG version are HYPRE_BoomerAMGSetNumFunctions HYPRE_BoomerAMGSetDofFunc and HYPRE_BoomerAMGSetNodal For best performance it might be necessary to set certain parameters which will affect both coarsening and interpolation One important parameter is the strong threshold which can be set using the function HYPRE_BoomerAMGSetStrongThreshold The default value is 0 25 which ap pears to be a good choice for 2 dimensional problems and the low complexity coarsening algorithms A better choice for 3 dimensional problems appears to be 0 5 if one
93. system and possibly avoid the expensive setup of a preconditioner if a system can be solved efficiently with a diagonally scaled Krylov solver e g a strongly diagonally dominant system It first uses a diagonally scaled Krylov solver which can be chosen by the user the default is conjugate gradient but one should use GMRES if the matrix of the linear system to be solved is nonsymmetric It monitors how fast the Krylov solver converges If there is not sufficient progress the algorithm switches to a preconditioned Krylov solver If used through the Struct interface the solver is called StructHybrid and can be used with the preconditioners SMG and PFMG default It is called ParCSRHybrid if used through the IJ inter face and is used here with BoomerAMG The user can determine the average convergence speed by setting a convergence tolerance 0 lt 0 lt 1 via the routine HY PRE_Struct HybridSetConvergenceTol HYPRE_StructParCSRHybridSetConvergenceTol The default setting is 0 9 a Vi 4 is monitored within the chosen Krylov solver where b Az is the i th residual Convergence is considered too slow when 1 loi pil max pi pi 1 The average convergence factor pi gt 0 6 3 When this condition is fulfilled the hybrid solver switches from a diagonally scaled Krylov solver to a preconditioned solver 6 8 BoomerAMG is a parallel implementation of the algebraic multigrid method
94. t b_S ex hypre_Vector bHYPRE_PCG PCG_solver bHYPRE_PCG_Apply PCG_solver b amp x ex bHYPRE_Vector_deleteRef b ex bHYPRE_StructVector_deleteRef b_S ex integer 8 b_S integer 8 b integer 8 x integer 8 PCG_solver call bHYPRE_StructVector_Create_f b_S ex call bHYPRE_Vector__cast_f b_S b ex call bHYPRE_PCG_Apply_f PCG_solver b x ierr ex call bHYPRE_Vector_deleteRef_f b ex call bHYPRE_StructVector_deleteRef_f b_S ex Other code not shown would set up the matrix provide it and parameters to the solver set up the vectors and so forth 8 9 THE HYPRE OBJECT STRUCTURE 71 8 9 The HYPRE Object Structure Even though it is written in C hypre is object oriented in its conceptual design The object structure of hypre is often visible for example in names of structs and functions It is even more visible in the Babel based interface than in the C only interface In both interfaces you can use hypre without any knowledge of its object structure or object oriented programming This section is for you if you have some basic knowledge of object oriented concepts and if you are curious about how they appear in hypre In this section we will touch on some of the more notable features and peculiarities of the object system as seen through the Babel based interface It is completely described elsewhere the object structure is precisely defined in the a file Interfaces idl written in SIDL whic
95. t level should be defined as a unique part There are two parts in this example part 0 is the global coarse grid and part 1 is the single refinement patch Note that the coarse unknowns underneath the refinement patch gray dots in Figure are not real physical unknowns the solution in this region is given by the values on the refinement patch In setting up the composite grid matrix for hypre the equations for these dummy unknowns should be uncoupled from the other unknowns this can easily be done by setting all off diagonal couplings to zero in this region 3 2 STRUCTURED ADAPTIVE MESH REFINEMENT w LS o LT Aa 1 7 HYPRE_SStructGraph graph HYPRE_SStructStencil c_stencil x_stencil y_stencil int c_var 0 x_var 1 y_var 2 int part Create the graph object HYPRE SStructGraphCreate MPI COMM WORLD grid amp graph Set the cell centered x face and y face stencils for each part for part 0 part lt 5 part HYPRE_SStructGraphSetStencil graph part c_var c_stencil HYPRE_SStructGraphSetStencil graph part x_var x_stencil HYPRE_SStructGraphSetStencil graph part y_var y_stenci
96. terface _ex And in Fortran you should declare it as follows integer 8 ex Parameters are set differently in the C only and Babel based interfaces In the C interface there is a different Set function for each parameter the parameter name is part of the function name In the Babel based interface there are just a few Set Parameter interfaces The parameter is one of the arguments For details see section or look in an example file MPI communicators are passed differently in the native form in the C only interface and as a language neutral object in the Babel based interface For details see section In two rare cases the Babel based interface of hypre provides output arrays in a special format See section B 10 if you need to deal with it With the Babel based interface in the C or Fortran languages you will occasionally need to explicitly cast an object between different data types You do this with a special cast function Moreover for every time you create or cast an object you need a corresponding call of a Destroy or deleteRef function See the sections 8 7 and 8 8 for details 8 4 Names and Conventions In the C only interface most hypre function names look like HYPRE ClassFunction where Class is a class name and Function is a conceptual function name The C Babel interface is similar most function names look like bHYPRE Class Function In the Fortran Babel interface most 8 5 PARAMETERS AND ERROR FLAGS 67 function names
97. the hypre library Associated documentation may be accessed by viewing the README htm1 file in that same directory e Papers Presentations etc Articles and presentations related to the hypre software library and the solvers available in the library are available from the hypre web page at http www 11nl gov CASC hypre e Mailing Lists There are two hypre mailing lists that can be subscribed to through the hypre web page at http www llnl gov CASC hypre 1 3 HOW TO GET STARTED 3 1 hypre announce hypre announce lists 11nl gov The development team uses this list to announce new general releases of hypre It cannot be posted to by users 2 hypre beta announce hypre beta announceQlists llnl gov The development team uses this list to announce new beta releases of hypre It cannot be posted to by users 1 3 How to get started 1 3 1 Installing hypre As previously noted on most systems hypre can be built by simply typing configure followed by make in the top level source directory For more detailed instructions read the INSTALL file provided with the hypre distribution or refer to the last chapter in this manual Note the following requirements e To run in parallel hypre requires an installation of MPI e Configuration of hypre with threads requires an implementation of OpenMP Currently only a subset of hypre is threaded e The hypre library currently does not support complex valued systems 1 3 2 Choosing a con
98. the coefficient data depending on the implementation The optional SetRowSizes and SetDiagOffdSizes routines mentioned later in this chapter and in the Reference Manual should be called before this step The SetValues routine sets matrix values for some number of rows nrows and some number of columns in each row ncols The actual row and column numbers of the matrix values to be set are given by rows and cols After the coefficients are set they can be added to with an AddTo routine Each process should set only those matrix values that it owns in the data distribution The Assemble routine is a collective call and finalizes the matrix assembly making the matrix ready to use The GetObject routine retrieves the built matrix object so that it can be passed on to hypre solvers that use the ParCSR internal storage format Note that this is not an expensive routine the matrix already exists in ParCSR storage format and the routine simply returns a handle or pointer to it Although we currently only support one underlying data storage format in the future several different formats may be supported 5 2 IJ VECTOR INTERFACE 33 One can preset the row sizes of the matrix in order to reduce the execution time for the matrix specification One can specify the total number of coefficients for each row the number of coefficients in the row that couple the diagonal unknown to Diag unknowns in the same processor domain and the
99. the discretization is symmetric only half of the off diagonal coefficients need to be stored To turn symmetric storage on the following line of code needs to be inserted somewhere between the Create and Initialize calls HYPRE_StructMatrixSetSymmetric A 1 The coefficients for the entire stencil can be passed in as before Note that symmetric storage may or may not actually be used depending on the underlying storage scheme Currently in hypre the Struct interface always uses symmetric storage To most efficiently utilize the Struct interface for symmetric matrices notice that only half of the off diagonal coefficients need to be set To do this for the example being considered we simply 2 5 SYMMETRIC MATRICES 15 need to redefine the 5 pt stencil of Section 2 2 to an appropriate 3 pt stencil then set matrix coefficients as in Section 2 3 for these three stencil elements only For example we could use the following stencil 0 0 1 0 2 3 This 3 pt stencil provides enough information to recover the full 5 pt stencil geometry and associated matrix coefficients 16 CHAPTER 2 STRUCTURED GRID SYSTEM INTERFACE STRUCT Chapter 3 Semi Structured Grid System Interface SStruct The SStruct interface is appropriate for applications with grids that are mostly but not entirely structured e g block structured grids see Figure 3 2 composite grids in structured adaptive mesh refinement AMR applic
100. their features are accessible through their common Solver interface But each has its own peculiarities and each takes a different set of parameters Most solvers are specific to one or two matrix classes An exception is the four Krylov solvers which will run with any matrix In contrast to the solvers matrices have almost nothing in common other than being operators ie a matrix can multiply a vector to compute another vector Once the storage scheme of the matrix is specified there is much to do such as specifying stencils index ranges and grids There are four different classes of matrix for four different storage schemes ParCSR structured and two kinds of semi structured matrix ParCSR matrices are generally used for unstructured meshes but do not incorporate the mesh A structured matrix includes a structured grid The semistruc tured matrix classes share the same interface with minimal modifications and both contain mesh information But their implementations e g matrix storage are different Conceptually there can be multiple views or ways of interfacing to a matrix Or one type of view can be used for more than one matrix class At present the Babel interface doesn t contain 72 CHAPTER 8 BABEL BASED INTERFACES l UParCSRMatrix 4 lJParVector class class 7 n Sk L9 2397 San Sisas Vector interface Operator interface A 2 5 M BoomerAMG Solver isa class
101. tive The global indexes allow hypre to discern how data is related spatially and how it is distributed across the parallel machine The basic component of the grid is a a collection of abstract cell centered indices in index space described by its lower and upper corner indices The scalar grid data is always associated with cell centers unlike the more general SStruct interface which allows data to be associated with box indices in several different ways Each process describes that portion of the grid that it owns one box at a time For example the global grid in Figure 2 1 can be described in terms of three boxes two owned by process 0 and one owned by process 1 F igureD 3 shows the code for setting up the grid on process 0 the code for process 1 is similar The icons at the top of the figure illustrate the result of the numbered lines of code The Create routine creates an empty 2D grid object that lives on the MPI COMM WORLD communicator The SetExtents routine adds a new box to the grid The Assemble routine is a collective call i e must be called on all processes from a common synchronization point and finalizes the grid assembly making the grid ready to use 2 1 SETTING UP THE STRUCT GRID HYPRE StructGrid grid int ndim 2 int ilower 2 3 1 0 1 int iupper 2 1 2 2 4 Create the grid object HYPRE_StructGrid
102. torization algorithm The original version of PILUT was done by Karypis and Kumar 13 in terms of the Cray SHMEM library The code was subsequently modified by the hypre team SHMEM was replaced by MPI some algorithmic changes were made and it was software engineered to be interoperable with several matrix implementations including hypre s ParCSR format PETSc s matrices and ISIS RowMatrix The algorithm produces an approximate factorization LU with the preconditioner M defined by M LU Note PILUT produces a nonsymmetric preconditioner even when the original matrix is sym metric Thus it is generally inappropriate for preconditioning symmetric methods such as Conju gate Gradient Parameters e SetMaxNonzerosPerRow int LFIL Default 20 Setthe maximum number of nonze ros to be retained in each row of L and U This parameter can be used to control the amount of memory that L and U occupy Generally the larger the value of LFIL the longer it takes to calculate the preconditioner and to apply the preconditioner and the larger the storage re quirements but this trades off versus a higher quality preconditioner that reduces the number of iterations 54 CHAPTER 6 SOLVERS AND PRECONDITIONERS e SetDropTolerance double tol Default 0 0001 Set the tolerance relative to the 2 norm of the row below which entries in L and U are automatically dropped PILUT first drops entries based on the drop tolerance and then retains t
103. uses the default coarsening al gorithm or CLJP However the choice of the strength threshold is problem dependent and therefore there could be better choices than the two suggested ones 6 9 AMS AMS Auxiliary space Maxwell Solver is a parallel unstructured Maxwell solver for edge finite element discretizations of the variational problem Find u E V a V xu V x v Bu v f v for all v 6 4 Here Vp is the lowest order Nedelec edge finite element space and a gt 0 and 8 gt 0 are scalar or SPD matrix coefficients AMS was designed to be scalable on problems with variable coefficients and allows for to be zero in part or the whole domain In either case the resulting problem is only semidefinite and for solvability the right hand side should be chosen to satisfy compatibility conditions AMS is based on the auxiliary space methods for definite Maxwell problems proposed in 9 For more details see 14 6 9 1 Overview Let A and b be the stiffness matrix and the load vector corresponding to 6 4 Then the resulting linear system of interest reads Ax b 6 5 The coefficients a and are naturally associated with the stiffness and mass terms of A Besides A and b AMS requires the following additional user input 44 CHAPTER 6 SOLVERS AND PRECONDITIONERS 1 The discrete gradient matrix G representing the edges of the mesh in terms of its vertices G has as many rows as the number of edges
104. v spaces Numer Math 2006 to appear 75 76 BIBLIOGRAPHY 10 Hysom and A Pothen Efficient parallel computation of ILU k preconditioners In Pro 11 12 13 14 15 16 17 18 19 20 21 22 23 ceedings of Supercomputing 99 ACM November 1999 Published on CDROM ISBN 1 58113 091 0 ACM Order 415990 IEEE Computer Society Press Order RS00197 D Hysom and A Pothen A scalable parallel algorithm for incomplete factor preconditioning SIAM J Sci Comput 22 6 2194 2215 2001 J Jones and B Lee A multigrid method for variable coefficient maxwell s equations SIAM J Sci Comput 27 1689 1708 2006 G Karypis and V Kumar Parallel threshold based ILU factorization Technical Report 061 University of Minnesota Department of Computer Science Army HPC Research Center Minneapolis MN 5455 1998 Tz V Kolev and P S Vassilevski Parallel H based auxiliary space AMG solver for curl problems Technical Report UCRL TR 222763 LLNL 2006 Tz V Kolev and P S Vassilevski Some experience with a H based auxiliary space AMG for H curl problems Technical Report UCRL TR 221841 LLNL 2006 5 F McCormick Multilevel Adaptive Methods for Partial Differential Equations volume 6 of Frontiers in Applied Mathematics SIAM Books Philadelphia 1989 J E Morel Randy M Roberts and Mikhail J Shashkov A local support operators diffusion discret
105. when using tens of thousands of processors Chapter 6 Solvers and Preconditioners There are several solvers available in hypre via different conceptual interfaces see Table 6 1 Note that there are a few additional solvers and preconditioners not mentioned in the table that can be used only through the FEI interface and are described in Paragraph 6 14 The procedure for setup and use of solvers and preconditioners is largely the same We will refer to them both as solvers in the sequel except when noted In normal usage the preconditioner is chosen and constructed before the solver and then handed to the solver as part of the solver s setup In the following we assume the most common usage pattern in which a single linear system is set up and then solved with a single righthand side We comment later on considerations for other usage patterns Setup 1 Pass to the solver the information defining the problem In the typical user cycle the user has passed this information into a matrix through one of the conceptual interfaces prior to setting up the solver In this situation the problem definition information is then passed to the solver by passing the constructed matrix into the solver As described before the matrix and solver must be compatible in that the matrix must provide the services needed by the solver Krylov solvers for example need only a matrix vector multiplication Most preconditioners on the other hand have additiona
106. xation is of nodal type all variables at a given point location are simultaneously solved for in the relaxation Although SysPFMG is implemented through the SStruct interface it can be used only for problems with one grid part Ideally the solver should handle any of the seven variable types cell node xface yface zface xedge yedge and zedge based However it has been completed only for cell based variables 6 4 SplitSolve SplitSolve is a parallel block Gauss Seidel solver for semi structured problems with multiple parts For problems with only one variable it can be viewed as a domain decomposition solver with no grid overlapping Consider a multiple part problem given by the linear system Az b Matrix A can be decom posed into a structured intra variable block diagonal component M and a component N consisting of the inter variable blocks and any unstructured connections between the parts SplitSolve per forms the iteration 1 M b Nay where M is a decoupled block diagonal V 1 1 cycle a separate cycle for each part and variable type There are two V cycle options SMG and PFMG 6 5 FAC FAC is a parallel fast adaptive composite grid solver for finite volume cell centred discretizations of smooth diffusion coefficient problems To be precise it is a FACx algorithm since the patch solves consist of only relaxation sweeps For details of the basic overall algorithms see 16 Algorithmic particularities
107. y used Krylov based iterative methods to be used in conjunction with its scalable preconditioners This includes methods for nonsymmetric systems such as GMRES and methods for symmetric matrices such as Conjugate Gradient e Intuitive grid centric interfaces obviate need for complicated data structures and provide access to advanced solvers hypre has made a major step forward in usability from earlier generations of sparse linear solver libraries in that users do not have to learn complicated sparse matrix data structures Instead hypre does the work of building these data structures for the user through a variety of conceptual interfaces each appropriate to 2 CHAPTER 1 INTRODUCTION different classes of users These include stencil based structured semi structured interfaces most appropriate for finite difference applications a finite element based unstructured inter face and a linear algebra based interface Each conceptual interface provides access to several solvers without the need to write new interface code e User options accommodate beginners through experts hypre allows a spectrum of expertise to be applied by users The beginning user can get up and running with a minimal amount of effort More expert users can take further control of the solution process through various parameters e Configuration options to suit your computing system hypre allows a simple and flexible installation on a wide variety of computing systems Users c

User's Manual - Computation - Lawrence Livermore National

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