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PETSc

PETSc. Portable, Extensible Toolkit for Scientific computing. Overview. Parallel & Sequential Object-oriented Available for virtually all UNIX platforms, as well as Windows 95/NT Flexible: many options for solver algorithms and parameters. Motivation.

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PETSc

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  1. PETSc Portable, Extensible Toolkit for Scientific computing

  2. Overview • Parallel & Sequential • Object-oriented • Available for virtually all UNIX platforms, as well as Windows 95/NT • Flexible: many options for solver algorithms and parameters

  3. Motivation • Developing parallel, non-trivial PDE solvers that deliver high performance is still difficult and requires months (or even years) of concentrated effort. • PETSc is a toolkit that can ease these difficulties and reduce the development time

  4. Introduction • A freely available and supported research code • Available via http://www.mcs.anl.gov/petsc • Free for everyone, including industrial users • Hyperlinked documentation and manual pages for all routines. Many tutorial-style examples • Support via email: petsc-maint@mcs.anl.gov • Usable from Fortran 77/90, C, and C++ • Portable to any parallel system supporting MPI, including • Tightly coupled systems • Cray T3E, SGI Origin, IBM SP, HP 9000, Sun Enterprise • Loosely coupled systems, e.g., networks of workstations • Compaq, HP, IBM, SGI, Sun, PCs running Linux or Windows

  5. History • PETSc history • Begun in September 1991 • Keep on development • Now: over 8,500 downloads since 1995 (versions 2.0 and 2.1)

  6. PETSc Concepts • How to specify the mathematics of the problem • Data objects - vectors, matrices • How to solve the problem • Solvers - linear, nonlinear, and time stepping (ODE) solvers • Parallel computing complications • Parallel data layout - structured and unstructured meshes

  7. PETSc PDE Application Codes ODE Integrators Visualization Nonlinear Solvers, Unconstrained Minimization Interface Linear Solvers Preconditioners + Krylov Methods Object-Oriented Matrices, Vectors, Indices Grid Management Profiling Interface Computation and Communication Kernels MPI, MPI-IO, BLAS, LAPACK Structure of PETSc

  8. PETSc Numerical Components Nonlinear Solvers Time Steppers(For ODE) Newton-based Methods Other Euler Backward Euler Pseudo Time Stepping Other Line Search Trust Region Krylov Subspace Methods GMRES CG CGS Bi-CG-STAB TFQMR Richardson Chebychev Other Preconditioners Additive Schwartz Block Jacobi Jacobi ILU ICC LU (Sequential only) Others Matrices Compressed Sparse Row (AIJ) Blocked Compressed Sparse Row (BAIJ) Block Diagonal (BDIAG) Dense Matrix-free Other

  9. What is not in PETSc? • Discretizations • Unstructured mesh generation and refinement tools • Load balancing tools • Sophisticated visualization capabilities

  10. BUT! PETSc has interface to external software that provides some of this functionality • Linear solvers (AMG BlockSolve95 LUSOL SPAI SuperLU) • Optimization software (TAO, Veltisto) • Mesh and discretization tools (overture, SAMRAI) • ODE solvers (PVODE) • Others (Matlab, ParMETIS)

  11. Flow of Control for PDE Solution Main Routine Timestepping Solvers (TS) Nonlinear Solvers (SNES) Linear Solvers (SLES) PETSc PC KSP Application Initialization Function Evaluation Jacobian Evaluation Post- Processing User code PETSc code

  12. A simple PETSc program #include “petsc.h”int main( int argc, char *argv[] ){ int rank; PetscInitialize(&argc,&argv,PETSC_NULL,PETSC_NULL); MPI_Comm_rank(PETSC_COMM_WORLD,&rank ); PetscSynchronizedPrintf(PETSC_COMM_WORLD, “Hello World from %d\n”,rank); PetscFinalize(); return 0;}

  13. Sparse Matrix Computation in PETSc • Variety Data Structures • Variety Preconditioners • Variety Iterative Solvers • Variety interface to External Softwares

  14. Data Structure for sparse matrix • Compressed Sparse Row • Blocked compressed sparse Row • Block Diagonal

  15. Example(BCRS) • Block Compressed Sparse Row NR: Number of blocks per side of the matrix NNZB: Number of Non-Zero Blocks NB: Number of elements per side of the Block N: Number of elements per side of the matrix

  16. Preconditioners • Incomplete LU factorization • Jacobi, Gauss-Seidel,SOR • Schwarz; within blocks use LU, ILU, SOR, etc.

  17. Solvers • Direct method (LU) • Krylov method (CG, GMRES, BiCGstab, CGS, QMR, …) • Non-Krylov iterative method (Jacobi, G-S, SOR) • Pre-conditioned Krylov method

  18. Interface to external solver Both Iterative and Director solvers • AMG • BlockSolve95 • LUSOL • SPAI • SuperLU

  19. Functions • Define the linear system (Ax=b) MatCreate(), MatSetValue(), VecCreate() • Create the Solver SLESCreate(), SLESSetOperators() • Solve System of Equations SLESSolve() • Clean up SLESDestroy()

  20. Example • Solve Ax = b • www.cs.ucsb.edu/~jh/example.cc

  21. ComparisonI: Iterative methods provided for different packages

  22. ComparisonII: Preconditioners provided by different packages

  23. ComparisonIII: Performance • Test Matrix:

  24. ComparisonIII: Performance • Feed these matrices to Aztec and PETSc • Generate a zero vector as initial guess and a vector of 1 as RHS • Use GMRES without preconditoners, max iteration = 500 • Run on 512-processor Cray T3E900

  25. ComparisonIII: Performance

  26. ComparisonIII: Performance

  27. ComparisonIII: Performance • Aztect suffers from setup varization • PETSc did not optimize partition algorithm( PE256 -> PE512, performance decrease). Communication takes more time than computation

  28. Conclusion • PETSc - Well designed and widely used - First MPI-based program to public - Good set of iterative methods and preconditioners - Good Support and excellent technical documents - Still under developement

  29. Thanks Comments ?

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