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#### 4.1 Examples

The code reported in Figure 1 shows how to set and apply the default multilevel preconditioner available in the real double precision version of AMG4PSBLAS (see Table 1 ). This preconditioner is chosen by simply specifying ’ML’ as the second argument of P%init (a call to P%set is not needed) and is applied with the CG solver provided by PSBLAS (the matrix of the system to be solved is assumed to be positive definite). As previously observed, the modules psb_base_mod, amg_prec_mod and psb_krylov_mod must be used by the example program.

The part of the code dealing with reading and assembling the sparse matrix and the right-hand side vector and the deallocation of the relevant data structures, performed through the PSBLAS routines for sparse matrix and vector management, is not reported here for the sake of conciseness. The complete code can be found in the example program file amg_dexample_ml.f90, in the directory samples/simple/fileread of the AMG4PSBLAS implementation (see Section 3.5 ). A sample test problem along with the relevant input data is available in samples/simple/fileread/runs. For details on the use of the PSBLAS routines, see the PSBLAS User’s Guide [ 20 ].

The setup and application of the default multilevel preconditioner for the real single precision and the complex, single and double precision, versions are obtained with straightforward modifications of the previous example (see Section 5 for details). If these versions are installed, the corresponding codes are available in samples/simple/fileread.

      use psb_base_mod
      use amg_prec_mod
      use psb_krylov_mod
    ... ...
    !
    ! sparse matrix
      type(psb_dspmat_type) :: A
    ! sparse matrix descriptor
      type(psb_desc_type)   :: desc_A
    ! preconditioner
      type(amg_dprec_type)  :: P
    ! right-hand side and solution vectors
      type(psb_d_vect_type) :: b, x
    ... ...
    !
    ! initialize the parallel environment
      call psb_init(ctxt)
      call psb_info(ctxt,iam,np)
    ... ...
    !
    ! read and assemble the spd matrix A and the right-hand side b
    ! using PSBLAS routines for sparse matrix / vector management
    ... ...
    !
    ! initialize the default multilevel preconditioner, i.e. V-cycle
    ! with basic smoothed aggregation, 1 hybrid forward/backward
    ! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
    ! solver
      call P%init(ctxt,’ML’,info)
    !
    ! build the preconditioner
      call P%hierarchy_build(A,desc_A,info)
      call P%smoothers_build(A,desc_A,info)
    !
    ! set the solver parameters and the initial guess
      ... ...
    !
    ! solve Ax=b with preconditioned FCG
      call psb_krylov(’FCG’,A,P,b,x,tol,desc_A,info)
      ... ...
    !
    ! deallocate the preconditioner
      call P%free(info)
    !
    ! deallocate other data structures
      ... ...
    !
    ! exit the parallel environment
      call psb_exit(ctxt)
      stop

Listing 1: setup and application of the default multilevel preconditioner (example 1).

Different versions of the multilevel preconditioner can be obtained by changing the default values of the preconditioner parameters. The code reported in Figure 2 shows how to set a V-cycle preconditioner which applies 1 block-Jacobi sweep as pre- and post-smoother, and solves the coarsest-level system with 8 block-Jacobi sweeps. Note that the ILU(0) factorization (plus triangular solve) is used as local solver for the block-Jacobi sweeps, since this is the default associated with block-Jacobi and set by P%init. Furthermore, specifying block-Jacobi as coarsest-level solver implies that the coarsest-level matrix is distributed among the processes. Figure 3 shows how to set a W-cycle preconditioner using the Coarsening based on Compatible Weighted Matching, aggregates of size at most 8 and smoothed prolongators. It applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother, and solves the coarsest-level system with the parallel flexible Conjugate Gradient method (KRM) coupled with the block-Jacobi preconditioner having ILU(0) on the blocks. Default parameters are used for stopping criterion of the coarsest solver. Note that, also in this case, specifying KRM as coarsest-level solver implies that the coarsest-level matrix is distributed among the processes.

The code fragments shown in Figures 2 and 3 are included in the example program file amg_dexample_ml.f90 too.

Finally, Figure 4 shows the setup of a one-level additive Schwarz preconditioner, i.e., RAS with overlap 2. Note also that a Krylov method different from CG must be used to solve the preconditioned system, since the preconditione in nonsymmetric. The corresponding example program is available in the file amg_dexample_1lev.f90.

For all the previous preconditioners, example programs where the sparse matrix and the right-hand side are generated by discretizing a PDE with Dirichlet boundary conditions are also available in the directory samples/simple/pdegen.

    ... ...
    ! build a V-cycle preconditioner with 1 block-Jacobi sweep (with
    ! ILU(0) on the blocks) as pre- and post-smoother, and 8  block-Jacobi
    ! sweeps (with ILU(0) on the blocks) as coarsest-level solver
      call P%init(ctxt,’ML’,info)
      call P%set(’SMOOTHER_TYPE’,’BJAC’,info)
      call P%set(’COARSE_SOLVE’,’BJAC’,info)
      call P%set(’COARSE_SWEEPS’,8,info)
      call P%hierarchy_build(A,desc_A,info)
      call P%smoothers_build(A,desc_A,info)
    ... ...

Listing 2: setup of a multilevel preconditioner based on the default decoupled coarsening

    ... ...
    ! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps
    ! as pre- and post-smoother, a distributed coarsest
    ! matrix, and MUMPS as coarsest-level solver
      call P%init(ctxt,’ML’,info)
      call P%set(’PAR_AGGR_ALG’,’COUPLED’,info)
      call P%set(’AGGR_TYPE’,’MATCHBOXP’,info)
      call P%set(’AGGR_SIZE’,8,info)
      call P%set(’ML_CYCLE’,’WCYCLE’,info)
      call P%set(’SMOOTHER_TYPE’,’FBGS’,info)
      call P%set(’SMOOTHER_SWEEPS’,2,info)
      call P%set(’COARSE_SOLVE’,’KRM’,info)
      call P%set(’COARSE_MAT’,’DIST’,info)
      call P%set(’KRM_METHOD’,’FCG’,info)
      call P%hierarchy_build(A,desc_A,info)
      call P%smoothers_build(A,desc_A,info)
    ... ...

Listing 3: setup of a multilevel preconditioner based on the coupled coarsening using weighted matching

    ... ...
    ! set RAS with overlap 2 and ILU(0) on the local blocks
      call P%init(ctxt,’AS’,info)
      call P%set(’SUB_OVR’,2,info)
      call P%bld(A,desc_A,info)
    ... ...
    ! solve Ax=b with preconditioned BiCGSTAB
      call psb_krylov(’BICGSTAB’,A,P,b,x,tol,desc_A,info)

Listing 4: setup of a one-level Schwarz preconditioner.

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