Chapter 1 parallel sparse solver - freeFEM.org

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20 CHAPTER 1. PARALLEL SPARSE SOLVER dparm[0] HIPS PREC: Relative residual norm: Default=1e-9 dparm[1] HIPS DROPTOL0: Numerical threshold in ILUT for interior domain (important : set 0.0 in HYBRID: Default=0.005) HIPS DROPTOL1 : Numerical threshold in ILUT for dparm[2] Schur preconditioner: Default=0.005 dparm[3] HIPS DROPTOLE : Numerical threshold for coupling between the interior level and Schur: Default 0.005 dparm[4] HIPS AMALG : Numerical threshold for coupling between the interior level and Schur: Default=0.005 dparm[5] HIPS DROPS CHUR : Numerical threshold for coupling between the interior level and Schur: Default=0.005 Table 1.12: Significations of dparams corresponding to HIPS interface 1.3.3 Interfacing with HYPRE HYPRE ( High Level Preconditionner) is a suite of parallel preconditionner developped at Lawrence Livermore National Lab [19] . They are two main classes of preconditionners developped in HYPRE: AMG (Algebraic MultiGrid) and Parasails (Parallel Sparse Approximate Inverse). Suppose we want to solve Ax = b. At the heart of AMG is a series of progressively coarser(smaller) representations of the matrix A. Given an approximation ˆx to the solution x, consider solving the residual equation Ae = r to find the error e, where r = b − A ˆx. A fundamental principle of AMG is that it is an algebraically smooth error. To reduce the algebraically smooth errors further, they need to be represented by a smaller defect equation (coarse grid residual equation) Acec = rc, which is cheaper to solve. After solving this coarse equation, the solution is then interpolated in fine grid represented here by matrix A. The quality of AMG depends on the choice of coarsening and interpolating operators. The sparse approximate inverse approximates the inverse of a matrix A by a sparse matrix M. A technical idea to construct matrix M is to minimize the Frobenuis norm of the residual matrix I − MA. For more details on this preconditioner technic see [29]. HYPRE implement three Krylov subspace solvers like GMRES, PCG and BiCGStab. Installation of HYPRE To install HYPRE, first download HYPRE package at [19], unpack and go to HYPRE/src source directory and do ./configure to configure Hypre. After this just type make all to create libHYPRE.a. Using HYPRE as interface to FreeFem++ Before calling HYPRE solver inside FreeFem++, you must compile file hypre FreeFem.cpp to create dynamic library hypre FreeFem.so. To do this, move to the directory src/solver of FreeFem++, edit file make f ile.inc to specify the following variables:
1.3. PARALLEL SPARSE ITERATIVE SOLVER 21 HYPRE DIR : Directory of HYPRE HYPRE INCLUDE = -I$(HYPRE DIR)src/hypre/include/ : Directory for header of HYPRE HYPRE LIB = -L$(HIPS DIR)/src/lib/ -lHYPRE : Hypre Librairy FREEFEM : FreeFem++ directory FREEFEM INCLUDE : FreeFem heaeder for sparse linear solver MET IS : METIS directory MET IS LIB : METIS librairy MPI : MPI directory MPI INCLUDE : MPI headers Like with pARMS, the calling of HIPS in FreeFem++ can be done in three manners. We will present only one example where user specify his parameters towards keywords lparams and dparams. Laplacian 3D solve with HYPRE Let us consider again 3D Laplacian example inside FreeFem++ package where after discretization we want to solve linear equation with Hypre. Example 1.6 is Laplacian3D using Hypre as linear solver. Example 1.6 is same than 1.5 so we just present line where we set some Hypre parameters. We first load Hypre solver at line line 2. At line 4 to 15 we specifie parameter to set to Hypre solver and in line 43 we set parameters to Hypre solver. It should be noted that the meaning of the entries of these vectors is different from those of Hips . In the case of HYPRE, the meaning of differents entries of vectors iparm and dparm are gived in tables 1.13 to 1.17. In Table ?? results of running example 1.6 on Clustter Paradent of Grid5000 are reported. We can see in this running example the efficiency of parallelism in particular when AMG are use as preconditionner. Example 1.6 (Laplacian3D.edp) 1: load "msh3" 2: load "hipre_FreeFem" // load librairie 3: int nn=10,iii; 4: int[int] iparm(20); 5: real[int] dparm(6); 6: for(iii=0;iii
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Page 3 and 4: 1.1. USING SPARSE PARALLEL SOLVER I
Page 5 and 6: 1.2. SPARSE DIRECT SOLVER 5 Install
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Page 9 and 10: 1.2. SPARSE DIRECT SOLVER 9 where P
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Page 25 and 26: 1.4. DOMAIN DECOMPOSITION 25 cator
Page 27 and 28: 1.4. DOMAIN DECOMPOSITION 27 mpiSca
Page 29 and 30: 1.4. DOMAIN DECOMPOSITION 29 29: mp
Page 31 and 32: Bibliography [1] P. R. Amestoy, I.

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