MueLu User's Guide
Contents
1. Teuchos RCP lt Tpetra CrsMatrix lt gt gt A create A here std string optionsFile muelu0ptions xml Teuchos RCP lt MueLu Tpetra0perator gt mueLuPreconditioner MueLu CreateTpetraPreconditioner A optionsFile The XML file contains multigrid options A typical file with MUELU parameters looks like the following lt ParameterList name MueLu gt lt Parameter name verbosity type string value low gt lt Parameter name max levels type int value 3 gt lt Parameter name coarse max size type int value 10 gt lt Parameter name multigrid algorithm type string value sa gt lt Damped Jacobi smoothing gt lt Parameter name smoother type type string value RELAXATION gt lt ParameterList name smoother params gt lt Parameter name relaxation type type string value Jacobi gt lt Parameter name relaxation sweeps type int value 1 gt lt Parameter name relaxation damping factor type double value 0 9 gt lt ParameterList gt lt Aggregation gt 19 lt Parameter name aggregation type type string value uncoupled gt lt Parameter name aggregation min agg size type int value 3 gt lt Parameter name aggregation max agg size type int value 9 gt lt ParameterList gt It defines a three level smoothed aggregation multigrid algorithm The aggregation size is between three and nine 2D 272. Support for the Epetra 64 bit version is planned 15 Modern object oriented software architecture MUELU is written completely in C as a modular object oriented multigrid framework which provides flexibility to combine and reuse existing components to develop novel multi grid methods Extensibility Due to its flexible design MUELU is an excellent toolkit for research on novel multigrid concepts Experienced multigrid users have full access to the underlying framework through an advanced XML based interface Expert users may use and extend the C API directly Integration with TRILINOS library As a package of TRILINOS MUELU is well integrated into the TRILINOS environment MUELU can be used with either the TPETRA or EPETRA 32 bit linear algebra stack It is templated on the local index global index scalar and compute node types This makes MUELU ready for future developments Broad range of supported platforms MUELU runs on wide variety of architectures from desktop workstations to parallel Linux clusters and supercomputers 10 Open source MUELU is freely available through a simplified BSD license see Appendix A 3 2 Configuration and Build MUELU has been compiled successfully under Linux with the following C compilers GNU versions 4 1 and later Intel versions 12 1 13 1 and clang versions 3 2 and later In the future we recommend using compilers supporting C 11 standard 3 2 1 Dependencies Req
3. lt Parameter name smoother type type string value CHEBYSHEV gt lt ParameterList gt changes the smoother for level 2 to be a polynomial smoother 5 2 Parameter validation By default MUELU validates the input parameter list A parameter that is misspelled is un known or has an incorrect value type will cause an exception to be thrown and execution to halt Spaces are important within a parameter s name Please separate words by just one space and make sure there are no leading or trailing spaces 25 The option print initial parameters prints the initial list given to the interpreter The option print unused parameters prints the list of unused parameters 5 3 General options Problem type Multigrid algorithm Block size Smoother unknown Poisson 2D Smoothed aggregation 1 Chebyshev Poisson 3D Smoothed aggregation 1 Chebyshev Elasticity 2D Smoothed aggregation 3 Chebyshev Elasticity 3D Smoothed aggregation 6 Chebyshev ConvectionDiffusion Petrov Galerkin AMG 1 Gauss Seidel MHD Additive Schwarz method with one level of overlap and ILU O as a subdo main solver Unsmoothed aggregation Table 5 1 Supported problem types means not set problem type verbosity number of equations max levels cycle type string Type of problem to be solved Possible values see Table 5 1 Default unknown string Control of the amount of printed information Possible values n
4. Copyright 2012 Sandia Corporation Under the terms of Contract DE AC04 94AL853000 with Sandia Corporation the U S Govern ment retains certain rights in this software Redistribution and use in source and binary forms with or without modification are permitted provided that the following conditions are met 1 Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer 2 Redistributions in binary form must reproduce the above copyright notice this list of con ditions and the following disclaimer in the documentation and or other materials provided with the distribution 3 Neither the name of the Corporation nor the names of the contributors may be used to endorse or promote products derived from this software without specific prior written permission THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION AS IS AND ANY EX PRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL SANDIA CORPORATION OR THE CONTRIBU TORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEQUENTIAL DAMAGES INCLUDING BUT NOT LIMITED TO PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE DATA OR PROFITS OR BUSI NESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE
5. Jonathan J Hu Christopher M Siefert Scalable Algorithms Computational Multiphysics Sandia National Laboratories Sandia National Laboratories Mailstop 9159 Mailstop 1322 P O Box 0969 P O Box 5800 Livermore CA 94551 0969 Albuquerque NM 87185 1322 jhu sandia gov csiefer sandia gov Raymond S Tuminaro Computational Mathematics Sandia National Laboratories Mailstop 9159 P O Box 0969 Livermore CA 94551 0969 rstumin Osandia gov Abstract This is the official user guide for MUELU multigrid library in Trilinos version 11 12 This guide provides an overview of MUELU its capabilities and instructions for new users who want to start using MUELU with a minimum of effort Detailed information is given on how to drive MUELU through its XML interface Links to more advanced use cases are given This guide gives information on how to achieve good parallel performance as well as how to introduce new algorithms Finally readers will find a comprehensive listing of available MUELU options Any options not documented in this manual should be considered strictly experimental Acknowledgment Many people have helped develop MUELU and or provided valuable feedback and we would like to acknowledge their contributions here Tom Benson Julian Cortial Eric Cyr Stefan Domino Travis Fisher Jeremie Gaidamour Axel Gerstenberger Chetan Jhurani Mark Hoemmen Paul Lin Eric Phipps Siva Rajamanickam Nico Schl mer and Paul Tsuji Contents Introd
6. repartition enable bool Rebalancing on off switch Default false 30 repartition repartition repartition repartition per proc repartition imbalance repartition repartition and R partitioner params start level min rows max remap parts rebalance P string Partitioning package to use Possible values zoltan ZOLTAN library zoltan2 ZOLTAN2 li brary Default zoltan2 ParameterList Partitioner parameters MUELU passes them directly to the appropriate package library int Minimum level to run partitioner MUELU does not rebalance levels finer than this one Default 2 int Minimum number of rows per processor If the actual number if smaller then rebalancing occurs De fault 800 double Maximum nonzero imbalance ratio If the actual number is larger the rebalancing occurs De fault 1 2 bool Postprocessing for partitioning to reduce data migration Default true bool Explicit rebalancing of R and P during the setup This speeds up the solve but slows down the setup phases Default true 5 7 Multigrid algorithm options multigrid algorithm semicoarsen rate coarsen sa damping factor string Multigrid method Possible values see Ta ble 5 5 Default sa int Rate at which to coarsen unknowns in the z direc tion Default 3 double Damping factor for smoothed aggregation Default 1 33 31 sa Cla
7. 3D nodes One sweep with a damped Jacobi method is used as a level smoother By default a direct solver is applied on the coarsest level A complete list of available parameters and valid parameter choices is given in 85 of this User s Guide Users can also construct a multigrid preconditioner using a provided ParameterList without accessing any files in the following manner Teuchos RCP lt Tpetra CrsMatrix lt gt gt A create A here Teuchos ParameterList paramlist paramList set verbosity low paramList set max levels 3 paramList set coarse max size 10 paramList set multigrid algorithm sa L Teuchos RCP lt MueLu Tpetra0perator gt mueLuPreconditioner MueLu CreateTpetraPreconditioner A paramList Besides the linear operator A we also need an initial guess vector for the solution X and a right hand side vector B for solving a linear system Teuchos RCP lt const Tpetra Map lt gt gt map A gt getDomainMap Create initial vectors Teuchos RCP lt Tpetra MultiVector lt gt gt B X X Teuchos rcp new Tpetra MultiVector lt gt map numrhs Belos MultiVecTraits lt gt MvRandom X B Teuchos rcp new Tpetra MultiVector lt gt map numrhs Belos OperatorTraits lt gt Apply A X B Belos MultiVecTraits lt gt MvInit X 0 0 To generate a dummy example the above code first declares two vectors Then a righ
8. 85 6 Try choosing rebalancing parameters so that you end up with one processor on the coarsest level for the direct solver this avoids additional communication 5 Enable implicit rebalancing of prolongators and restrictors repartition rebalance P and R 24 Chapter 5 MUELU options In this section we report the complete list of MUELU input parameters It is important to notice however that MUELU relies on other TRILINOS packages to provide support for some of its algorithms For instance IFPACK IFPACK2 provide standard smoothers like Jacobi Gauss Seidel or Chebyshev while AMESOS AMESOS2 provide access to direct solvers The parameters affecting the behavior of the algorithms in those packages are simply passed by MUELU to a routine from the corresponding library Please consult corresponding packages documentation for a full list of supported algorithms and corresponding parameters 5 1 Using parameters on individual levels Some of the parameters that affect the preconditioner can in principle be different from level to level By default parameters affect all levels in a multigrid hierarchy The settings on a particular levels can be changed by using level sublists A level sublist is a ParameterList sublist with the name level XX where XX is the level number The parameter names in the sublist do not require any modifications For example the following fragment of code lt ParameterList name level 2 gt
9. Az b u end if end function In the multigrid iteration in Algorithm 1 the S s and 2 s are called pre smoothers and post smoothers They are approximate solvers e g symmetric Gauss Seidel with the subscript m denoting the number of applications of the approximate solution method The purpose of a smoother is to quickly reduce certain error modes in the approximate solution on a level k For symmetric problems the pre and post smoothers should be chosen to maintain symmetry e g forward Gauss Seidel for the pre smoother and backward Gauss Seidel for the post smoother The P s are interpolation matrices that transfer solutions from coarse levels to finer levels The Rgs are restriction matrices that restrict a fine level solution to a coarser level In a geometric multigrid P s and Rg s are determined by the application whereas in an algebraic multigrid they are automatically generated For symmetric problems typically R Pr For nonsymmetric prob lems this is not necessarily true The A s are the coarse level problems and are generated through 13 a Galerkin triple matrix product Please note that the algebraic multigrid algorithms implemented in MUELU generate the grid transfers P automatically and the coarse problems Az via a sparse triple matrix product TRILINOS provides a wide selection of smoothers and direct solvers for use in MUELU through the IFPACK IFPACK2 AMESOS and AMESOS2 packages see 8
10. the dependencies of MUELU both required and optional If an optional depen dency is not present the tests requiring that dependency are not built AMESOS AMESOS2 are necessary if one wants to use a sparse direct solve on the coarsest level ZOLTAN ZOLTAN2 are necessary if one wants to use matrix rebalancing in parallel runs see 84 AZTECOO BELOS are necessary if one wants to test MUELU as a preconditioner instead of a solver MUELU has also been successfully tested with SuperLU 4 1 and SuperLU 4 2 Some packages that MUELU depends on may come with additional requirements for third party libraries which are not listed here as explicit dependencies of MUELU It is highly recommended to install ParMetis 3 1 1 or newer for ZOLTAN ParMetis 4 0 x for ZOLTAN2 and SuperLU 4 1 or newer for AMESOS AMESOS2 3 2 2 Configuration The preferred way to configure and build MUELU is to do that outside of the source direc tory Here we provide a sample configure script that will enable MUELU and all of its optional dependencies export TRILINOS_HOME path to your Trilinos source directory cmake D BUILD_SHARED_LIBS BOOL ON D CMAKE_BUILD_TYPE STRING RELEASE D CMAKE_CXX_FLAGS STRING g D Trilinos_ENABLE_EXPLICIT_INSTANTIATION BOOL ON D Trilinos_ENABLE_TESTS BOOL 0FF D Trilinos_ENABLE_EXAMPLES BOOL OFF D Trilinos_ENABLE_MueLu BOOL ON D MueLu_ENABLE_TESTS STRING ON D MueLu_ENABLE_EXAMPLES STRING ON
11. 5 14 Chapter 3 Getting Started This section is meant to get you using MUELU as quickly as possible 83 gives a summary of MUELU s design 83 2 lists MUELU s dependencies on other TRILINOS libraries and provides a sample cmake configuration line Finally code examples using the XML interface are given in 53 3 3 1 Overview of MUELU MUELU is an extensible algebraic multigrid AMG library that is part of the TRILINOS project MUELU works with EPETRA 32 bit version and TPETRA matrix types The library is written in C and allows for different ordinal index and scalar types MUELU is designed to be efficient on many different computer architectures from workstations to supercomputers relying on MPI X principle where X can be threading or CUDA MUELU provides a number of different multigrid algorithms 1 smoothed aggregation AMG for Poisson like and elasticity problems 2 Petrov Galerkin aggregation AMG for convection diffusion problems 3 energy minimizing AMG 4 aggregation based AMG for problems arising from the eddy current formulation of Maxwell s equations MUELU s software design allows for the rapid introduction of new multigrid algorithms The most important features of MUELU can be summarized as Easy to use interface MUELU has a user friendly parameter input deck which covers most important use cases Reasonable defaults are provided for common problem types see Table 5 1
12. D TPL_ENABLE_BLAS BOOL 0N D TPL_ENABLE_MPI BOOL ON TRILINOS_HOME More configure examples can be found in Trilinos sampleScripts For more information on configuring see the TRILINOS Cmake Quickstart guide 1 3 3 Examples in code The most commonly used scenario involving MUELU is using a multigrid preconditioner in side an iterative linear solver In TRILINOS a user has a choice between EPETRA and TPETRA 18 for the underlying linear algebra library Important Krylov subspace methods such as precondi tioned CG and GMRES are provided in TRILINOS packages AZTECOO EPETRA and BELOS EPETRA TPETRA At this point we assume that the reader is comfortable with TEUCHOS referenced counted pointers RCPs for memory management an introduction to RCPs can be found in 3 and the Teuchos Parameterlist class 13 3 3 1 MUELU as a preconditioner within BELOS The following code shows the basic steps of how to use a MUELU multigrid preconditioner with TPETRA linear algebra library and with a linear solver from BELOS To keep the example short and clear we skip the template parameters and focus on the algorithmic outline of setting up a linear solver For further details a user may refer to the examples and test directories First we create the MUELU multigrid preconditioner It can be done in a few ways For in stance multigrid parameters can be read from an XML file e g mueluOptions xml in the example below
13. J Schroder and R Tuminaro A general interpolation strategy for algebraic multi grid using energy minimization SIAM Journal on Scientific Computing 33 2 966 991 2011 Marzio Sala and Raymond S Tuminaro A new Petrov Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems SIAM Journal on Scientific Computing 31 1 143 166 2008 35 13 Heidi Thornquist Roscoe A Bartlett Mark Hoemmen Christopher Baker and Michael Her oux Teuchos Trilinos tools library http trilinos org packages teuchos 2014 14 Ulrich Trottenberg Cornelis Oosterlee and Anton Sch ller Multigrid Elsevier Academic Press 2001 15 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 16 Tobias A Wiesner Flexible Aggregation based Algebraic Multigrid Methods for Contact and Flow Problems PhD thesis 2014 17 Tobias A Wiesner Michael W Gee Andrey Prokopenko and Jonathan J Hu The MueLu tutorial http trilinos org packages muelu muelu tutorial 2014 SAND2014 18624R 18 Tobias A Wiesner Raymond S Tuminaro Wolfgang A Wall and Michael W Gee Multi grid transfers for nonsymmetric systems based on Schur complements and Galerkin projec tions Numerical Linear Algebra with Applications 21 3 415 438 2014 36 Appendix A Copyright and License MueLu A package for multigrid based preconditioning
14. MueLu Epetra0perator gt mueLuPreconditioner std string optionsFile mueluOptions xml mueLuPreconditioner MueLu CreateEpetraPreconditioner A optionsFile MUELU parameters are generally Epetra Tpetra agnostic thus the XML parameter file could be the same as 3 3 1 Furthermore we assume that a right hand side vector and a solution vector with the initial guess are defined Teuchos RCP lt const Epetra_Map gt map A gt DomainMap Teuchos RCP lt Epetra_Vector gt B Teuchos rcp new Epetra_Vector map Teuchos RCP lt Epetra_Vector gt X Teuchos rcp new Epetra_Vector map X gt PutScalar 0 0 Then an Epetra_LinearProblem can be defined 21 Epetra_LinearProblem epetraProblem A get X get B getO The following code constructs an AZTECOO CG solver Aztec00 aztecSolver epetraProblem aztecSolver SetAztecOption AZ_solver AZ_cg aztecSolver SetPrecOperator mueLuPreconditioner get Finally the linear system is solved int maxlts 100 double tol 1e 10 aztecSolver Iterate maxIts tol 3 3 3 Further remarks This section is only meant to give a brief introduction on how to use MUELU as a precon ditioner within the TRILINOS packages for iterative solvers There are other more complicated ways to use MUELU as a preconditioner for BELOS and AZTECOO through the XPETRA inter face Of course MUELU can also work as st
15. OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE 37 DISTRIBUTION Rh A a gt gt ed Tobias Wiesner Institute for Computational Mechanics Technische Universit t Miinchen Boltzmanstra e 15 85747 Garching Germany MS 1320 MS 1318 MS 1320 MS 1320 MS 1318 MS 1322 MS 0899 Michael Heroux 1446 Robert Hoekstra 1446 Mark Hoemmen 1446 Paul Lin 1446 Andrey Prokopenko 1426 Christopher Siefert 1443 Technical Library 9536 electronic copy 38 DISTRIBUTION p gt pm MS 9159 MS 9159 MS 9159 MS 0899 Jonathan Hu 1426 Paul Tsuji 1442 Raymond Tuminaro 1442 Technical Library 8944 electronic copy 39 40 v1 39 Ah Sandia National Laboratories
16. SANDIA REPORT SAND2014 18874 Unlimited Release Printed October 2014 MueLu User s Guide 1 0 Trilinos version 11 12 Andrey Prokopenko Jonathan J Hu Tobias A Wiesner Christopher M Siefert Raymond S Tuminaro Prepared by Sandia National Laboratories Albuquerque New Mexico 87185 and Livermore California 94550 Sandia National Laboratories is a multi program laboratory managed and operated by Sandia Corporation a wholly owned subsidiary of Lockheed Martin Corporation for the U S Department of Energy s National Nuclear Security Administration under contract DE AC04 94AL85000 Approved for public release further dissemination unlimited fh Sandia National Laboratories Issued by Sandia National Laboratories operated for the United States Department of Energy by Sandia Corporation NOTICE This report was prepared as an account of work sponsored by an agency ofthe United States Government Neither the United States Government nor any agency thereof nor any of their employees nor any of their contractors subcontractors or their employees make any warranty express or implied or assume any legal liability or responsibility for the accuracy completeness or usefulness of any information apparatus product or process disclosed or rep resent that its use would not infringe privately owned rights Reference herein to any specific commercial product process or service by trade name trademark manufacturer or other
17. andalone multigrid solver For more information on these topics the reader may refer to the examples and tests in the MUELU source directory Trilinos packages muelu as well as to the MUELU tutorial 17 For in depth details of MUELU applied to multiphysics problems please see 16 22 Chapter 4 Performance tips In practice it can be very challenging to find an appropriate set of multigrid parameters for a specific problem especially if few details are known about the underlying linear system In this Chapter we provide some advice for improving multigrid performance For optimizing multigrid parameters it is highly recommended to set the verbosity to high or extreme for MUELU to output more information e g for the effect of the chosen parameters to the aggregation and coarsening process Some general advice e Choose appropriate iterative linear solver e g GMRES for non symmetric problems e Start with the recommended settings for particular problem types See Table 5 1 e Choose reasonable basic multigrid parameters see 5 3 including maximum number of multigrid levels max levels and maximum allowed coarse size of the problem coarse max size Take fine level problem size and sparsity pattern into account for a reasonable choice of these parameters e Select an appropriate transfer operator strategy see 85 7 For symmetric problems you should start with smoothed aggregation multigrid For non symmetric pr
18. ed direct solvers provided by AME coarse type KLU x KLU2 SuperLU x SuperLU_dist x Umfpack x Mumps x SOS AMESOS2 smoother post type smoother params smoother pre params string Post smoother type Possible values see Ta ble 5 2 Default RELAXATION ParameterList Smoother parameters For standard smoothers MUELU passes them directly to the appro priate package library ParameterList Pre smoother parameters For stan dard smoothers MUELU passes them directly to the ap propriate package library 28 smoother smoother smoother smoother coarse coarse coarse coarse post params overlap pre overlap post overlap max size type params overlap ParameterList Post smoother parameters For stan dard smoothers MUELU passes them directly to the ap propriate package library int Smoother subdomain overlap Default 0 int Pre smoother subdomain overlap Default 0 int Post smoother subdomain overlap Default 0 int Maximum dimension of a coarse grid MUELU will stop coarsening once it is achieved Default 2000 string Coarse solver Possible values see Table Default SuperLU ParameterList Coarse solver parameters MUELU passes them directly to the appropriate package library int Coarse solver subdomain overlap Default 0 5 5 Aggregation options aggregation type aggregation ordering string Aggregation scheme P
19. in num reuse int Number of iterations to minimize the reused pro iterations longator energy in energy minimization Default 1 32 emin pattern emin pattern order string Sparsity pattern to use for energy minimiza tion Possible values AkPtent Default AkPtent int Matrix order for the AkPtent pattern De fault 1 5 8 Miscellaneous options export data print initial parameters print unused parameters transpose use implicit ParameterList Exporting a subset of the hierarchy data in a file Currently the list can contain any of three parameter names A P R of type string and value levels separated by commas A matrix with a name X is saved in three MatrixMarket files a data is saved in X_level mm b its row map is saved in rowmap_X_level mm c its column map is saved in colmap_X_level mm bool Print parameters provided for a hierarchy con struction Default true bool Print parameters unused during a hierarchy con struction Default true bool Use implicit transpose for the restriction opera tor Default false 33 34 References 1 2 10 11 12 Trilinos CMake Quickstart http trilinos org build_instructions html 2014 Patrick R Amestoy Iain S Duff Jean Yves L Excellent and Jacko Koster MUMPS a general purpose distributed memory sparse solver In Applied Parallel Computing New Paradigms f
20. ives an overview of MUELU s capabilities If you are looking for a tutorial please refer to the MUELU tutorial in muelu doc Tutorial see also 17 New users should start with 83 It strives to give the new user all the information he she might need to begin using MUELU quickly Users interested in performance especially in parallel context should refer to 84 Users looking for a particular option should consult 85 containing a complete set of supported options in MUELU If you find any errors or omissions in this guide have comments or suggestions or would like to contribute to MUELU development please contact the MUELU users list or developers list 11 12 Chanter 2 Multigrid background Here we provide a brief multigrid introduction see 5 or 14 for more information A multigrid solver tries to approximate the original problem of interest with a sequence of smaller coarser problems The solutions from the coarser problems are combined in order to acceler ate convergence of the original fine problem on the finest grid A simple multilevel iteration is illustrated in Algorithm 1 Algorithm 1 V cycle multigrid with N levels to solve Ax b Ag A function MULTILEVEL A b u k Solve A u b k is current grid level u S Az b u if k 4 N 1 then P determine_interpolant Az Ry determine_restrictor A Tk 1 KLD Ayu Axt RrArPk v 0 MULTILEVEL Ar 1 T V k 1 u u Pv u S2
21. ng package without any examination of their content Please consult the documentation of the corresponding packages for a list of possible values By default MUELU uses one sweep of symmetric Gauss Seidel for both pre and post smoothing and SuperLU for coarse system solver smoother pre or post string Pre and post smoother combination Possi ble values pre only pre smoother post only post smoother both both pre and post smoothers none no smoothing Default both smoother type string Smoother type Possible values see Table 5 2 Default RELAXATION smoother pre type string Pre smoother type Possible values see Ta ble 5 2 Default RELAXATION 27 smoother type RELAXATION CHEBYSHEV ILUT RILUK Point relaxation smoothers including Jacobi Gauss Seidel symmetric Gauss Seidel etc The exact smoother is cho sen by specifying relaxation type parameter in the smoother params sublist Table 5 2 PACK IFPACK 2 Chebyshev polynomial smoother Local processor based incomplete factorization methods Commonly used smoothers provided by IF Note that these smoothers can also be used as coarse grid solvers AMESOS AMESOS2 Default AMESOS solver 7 x Default AMESOS2 solver 4 x Third party serial sparse direct solver 9 x Third party parallel sparse direct solver 9 Third party solver 6 Third party solver 2 Table 5 3 Commonly us
22. oblems a Petrov Galerkin smoothed aggregation method is a good starting point though smoothed aggrega tion may also perform well e Enable implicit restrictor construction transpose use implicit for symmetric prob lems e Find good level smoothers see 5 4 If a problem is symmetric positive definite choose a smoother with a matrix vector computational kernel such as the Chebyshev polynomial smoother If you are using relaxation smoothers we recommend starting with optimizing the damping parameter Once you have found a good damping parameter for your problem you can increase the number of smoothing iterations e Adjust aggregation parameters if you experience bad coarsening ratios see 5 5 Particu larly try adjusting the minimum aggregation min agg size and maximum aggregation max agg size aggregation parameters For a 2D 3D isotropic problem on a regular mesh the aggregate size should be about 9 27 nodes per aggregate 23 e Replace a direct solver with an iterative method coarse type if your coarse level solu tion becomes too expensive see 85 4 Some advice for parallel runs include 1 Enable matrix rebalancing when running in parallel repartition enable 2 Use smoothers invariant to the number of processors such as polynomial smoothing if possible 3 Use uncoupled aggregation instead of coupled as later requires significantly more com munication 4 Adjust rebalancing parameters see
23. one low medium high extreme Default high int Number of PDE equations at each grid node Only constant block size is considered Default 1 int Maximum number of levels in a hierarchy De fault 10 string Multigrid cycle type Possible values V W Default V 26 problem symmetric bool Symmetry of a problem This setting affects the construction of a restrictor If set to true the restric tor is set to be the transpose of a prolongator If set to false underlying multigrid algorithm makes the deci sion Default true 5 4 Smoothing and coarse solver options MUELU relies on other TRILINOS packages to provide level smoothers and coarse solvers IFPACK and IFPACK2 provide standard smoothers see Table 5 2 and AMESOS and AMESOS2 provide direct solvers see Table 5 3 Note that it is completely possible to use any level smoother as a direct solver MUELU checks parameters smoother type and coarse type to determine e what package to use 1 e is it a smoother or a direct solver e possibly transform a smoother type IFPACK and IFPACK2 use different smoother type names e g point relaxation stand alone vs RELAXATION MUELU tries to follow IFPACK2 notation for smoother types Please consult IFPACK2 documentation 8 for more information The parameter lists smoother params and coarse params are passed directly to the corre spondi
24. or HPC in Industry and Academia pages 121 130 Springer 2001 Roscoe A Bartlett Teuchos RCP beginners guide Technical Report SAND2004 3268 Sandia National Labs 2010 Eric Bavier Mark Hoemmen Sivasankaran Rajamanickam and Heidi Thornquist Amesos2 and Belos Direct and iterative solvers for large sparse linear systems Scientific Program ming 20 3 241 255 2012 William L Briggs Van Emden Henson and Steve F McCormick A multigrid tutorial SIAM 2nd edition 2000 Timothy A Davis Algorithm 832 UMFPACK v4 3 an unsymmetric pattern multifrontal method ACM Transactions on Mathematical Software TOMS 30 2 196 199 2004 Timothy A Davis and Ekanathan Palamadai Natarajan Algorithm 907 KLU a direct sparse solver for circuit simulation problems ACM Transactions on Mathematical Software TOMS 37 3 36 2010 Mark Hoemmen Jonathan J Hu and Christopher Siefert Ifpack2 incomplete factoriza tions relaxations and domain decomposition library http trilinos org packages ifpack2 2014 Xiaoye S Li James W Demmel John R Gilbert Laura Grigori Meiyue Shao and Ichitaro Yamazaki SuperLU Users Guide 2011 Paul Lin Matthew Bettencourt Stefan Domino Travis Fisher Mark Hoemmen Jonathan J Hu Eric Phipps Andrey Prokopenko Sivasankaran Rajamanickam Christopher Siefert and Stephen Kennon Towards extreme scale simulations for low Mach fluids with second generation Trilinos To appear L Olson
25. ossible values see Ta ble 5 4 Default uncoupled string Node ordering strategy Possible values natu ral local index order graph filtered graph breadth first order random random local index order De fault natural 29 uncoupled coupled aggregation aggregation aggregation size aggregation size aggregation threshold aggregation Attempts to construct aggregates of optimal size 3 nodes in d dimensions Each process works independently and aggregates cannot span multiple processes Attempts to construct aggregates of optimal size 3 nodes in d dimensions Aggregates are allowed to cross proces sor boundaries Use carefully If unsure use uncoupled instead Table 5 4 Available coarsening schemes drop scheme drop tol min agg max agg Dirichlet export visualization data string Connectivity dropping scheme for a graph used in aggregation Possible values classical distance laplacian Default classical double Connectivity dropping threshold for a graph used in aggregation Default 0 0 int Minimum size of an aggregate Default 2 int Maximum size of an aggregate 1 means unlim ited Default 1 double Threshold for determining whether entries are zero during Dirichlet row detection Default 0 0 bool Export data for visualization post processing Default false 5 6 Rebalancing options
26. ssic smoothed aggregation 15 unsmoothed Aggregation based same as sa but without damped Jacobi pro longator improvement step pg Prolongator smoothing using A restriction smoothing using A local damping factors 12 emin Constrained minimization of energy in basis functions of grid transfer operator 18 11 Table 5 5 Available multigrid algorithms for generating grid transfer matrices sa use filtered matrix bool Matrix to use for smoothing the tentative prolon gator The two options are to use the original matrix and to use the filtered matrix with filtering based on fil tered graph used for aggregation Default true filtered matrix use bool Lump add to diagonal dropped entries during lumping the construction of a filtered matrix This allows user to preserve constant nullspace Default true filtered matrix reuse bool Skip eigenvalue calculation during the construc eigenvalue tion of a filtered matrix by reusing eigenvalue estimate from the original matrix This allows us to skip heavy computation but may lead to poorer convergence De fault true emin iterative method string Iterative method to use for energy minimiza tion of initial prolongator in energy minimization Pos sible values cg conjugate gradient sd steepest descent Default cg emin num iterations int Number of iterations to minimize initial prolon gator energy in energy minimization Default 2 em
27. t hand side vector is calculated as the matrix vector product of a random vector with the operator A Finally an initial guess is initialized with zeros Then one can define a Belos LinearProblem object where the mueLuPreconditioner is used for left preconditioning Belos LinearProblem lt gt problem A X B problem gt setLeftPrec mueLuPreconditioner 20 bool set problem setProblem Next we set up a BELOS solver using some basic parameters Teuchos ParameterList belosList belosList set Block Size 1 belosList set Maximum Iterations 100 belosList set Convergence Tolerance 1e 10 belosList set Output Frequency 1 belosList set Verbosity Belos TimingDetails Belos FinalSummary Belos BlockCGSolMgr lt gt solver rcp amp problem false rcp amp belosList false Finally we solve the system Belos ReturnType ret solver solve 3 3 2 MUELU as a preconditioner for AZTECOO For EPETRA users have two library options BELOS recommended and AZTECOO AZTECOO and BELOS both provide fast and mature implementations of common iterative Krylov linear solvers BELOS has additional capabilities such as Krylov subspace recycling and tall skinny QR Constructing a MUELU preconditioner for Epetra operators is done in a similar manner to Tpetra Teuchos RCP lt Epetra_CrsMatrix gt A create A here Teuchos RCP lt
28. uction Multigrid background Getting Started Sl TAI OF TT ua eg 3 2 Configuration and Build an a a EA Dependencies ehe ee ee aa 322 o AAA Fe AM II ri EEE ae 3 3 1 MUELU as a preconditioner within BELOS 00 000 ee 332 MUELU as a preconditioner for AZTECOO is 3 33 UCI TONITE SI E A RA Performance tips MUELU options 5 1 Using parameters on individual levels 2 e EA DONE aan arar ac bee aaa oa ero 5 4 Smoothing and coarse solver Opti0NS oooooocroorcocoor rr 5 3 ao o as ee as 56 o ok na ann re han un ae ah 5 7 Multigrid algorithm opliens 2 22 4502 sheen aneura rera 11 13 15 15 16 16 18 18 19 21 22 23 25 5 8 Miscellaneous options References Appendix A Copyright and License 37 List of Figures List of Tables 3 1 Sl 32 K 5 4 3 MUELU s required and optional dependencies subdivided by whether a depen dency is that of the MUELU library itself Library or of some MUELU test Test TI AAA AA A TIRA AAA AA ARA AA 17 Supported problem types means not set ae ann 26 Commonly used smoothers provided by IFPACK IFPACK2 Note that these smoothers can also be used as coarse grid IE eo 28 Commonly used direct solvers provided by AMESOS AMESOS2 28 Available coarsening schemes xssun san aan RARA cha 30 Available multigrid algorithms for generating grid transfer matrices 32 10 Chapter 1 Introduction This guide g
29. uired Dependencies MUELU requires that TEUCHOS and either EPETRA IFPACK or TPETRA IFPACK2 are enabled Recommended Dependencies We strongly recommend that you enable the following TRILINOS libraries along with MUELU e EPETRA stack AZTECOO EPETRA AMESOS IFPACK ISORROPIA GALERI ZOLTAN 16 e TPETRA stack AMESOS2 BELOS GALERI IFPACK2 TPETRA ZOLTAN2 Tutorial Dependencies In order to run the MUELU Tutorial 17 located in muelu doc Tutorial MUELU must be configured with the following dependencies enabled AZTECOO AMESOS AMESOS2 BELOS EPETRA IFPACK IFPACK2 ISORROPIA GALERI TPETRA ZOLTAN ZOLTAN2 Note that the MUELU tutorial 17 comes with a VirtualBox image with a pre installed Linux and TRILINOS In this way a user can immediately begin experimenting with MUELU without having to install the TRILINOS libraries Therefore 1t is an ideal starting point to get in touch with MUELU Complete List of Direct Dependencies Dependency Required Optional Library Testing Library Testing AMESOS x AMESOS2 X AZTECOO z BELOS EPETRA x gt IFPACK Se e IFPACK2 S ISORROPIA Y y GALERI E KOKKOSCLASSIC x TEUCHOS x x TPETRA x XPETRA x X ZOLTAN a ZOLTAN2 2 E Boost 2 BLAS x gt LAPACK x x MPI x w Table 3 1 MUELU s required and optional dependencies subdi vided by whether a dependency is that of the MUELU library itself Library or of some MUELU test Testing 17 Table 3 1 lists
30. wise does not necessarily constitute or imply its endorsement recommendation or favoring by the United States Government any agency thereof or any of their contractors or subcontractors The views and opinions expressed herein do not necessarily state or reflect those of the United States Government any agency thereof or any of their contractors Printed in the United States of America This report has been reproduced directly from the best available copy Available to DOE and DOE contractors from U S Department of Energy Office of Scientific and Technical Information P O Box 62 Oak Ridge TN 37831 Telephone 865 576 8401 Facsimile 865 576 5728 E Mail reports adonis osti gov Online ordering http www osti gov bridge Available to the public from U S Department of Commerce National Technical Information Service 5285 Port Royal Rd Springfield VA 22161 Telephone 800 553 6847 Facsimile 703 605 6900 E Mail orders ntis fedworld gov Online ordering http www ntis gov help ordermethods asp loc 7 4 O online SAND2014 18874 Unlimited Release Printed October 2014 MueLu User s Guide 1 0 Trilinos version 11 12 Andrey Prokopenko Tobias Wiesner Scalable Algorithms Institute for Computational Mechanics Sandia National Laboratories Technische Universit t M nchen Mailstop 1318 Boltzmanstra e 15 P O Box 5800 85747 Garching Germany Albuquerque NM 87185 1318 wiesner Inm mw tum de aprokop sandia gov
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