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Examples

The code reported in Figure 2 shows how to set and apply the default multi-level preconditioner available in the real double precision version of MLD2P4 (see Table 1). This preconditioner is chosen by simply specifying `'ML'` as second argument of `mld_precinit` (a call to `mld_precset` is not needed) and is applied with the BiCGSTAB solver provided by PSBLAS. As previously observed, the modules `psb_base_mod`, `mld_prec_mod` and `psb_krylov_mod` must be used by the example program.

The part of the code concerning the reading and assembling of the sparse matrix and the right-hand side vector, performed through the PSBLAS routines for sparse matrix and vector management, is not reported here for brevity; the statements concerning the deallocation of the PSBLAS data structure are neglected too. The complete code can be found in the example program file `mld_dexample_ml.f90`, in the directory `examples/fileread` of the MLD2P4 tree (see Section 3.5). For details on the use of the PSBLAS routines, see the PSBLAS User's Guide [15].

The setup and application of the default multi-level preconditioners for the real single precision and the complex, single and double precision, versions are obtained with straightforward modifications of the previous example (see Section 6 for details). If these versions are installed, the corresponding Fortran 95 codes are available in `examples/fileread/`.

Figure 2: Setup and application of the default multi-level Schwarz preconditioner.
      use psb_base_mod
      use mld_prec_mod
      use psb_krylov_mod
    ... ...
    !
    ! sparse matrix
      type(psb_dspmat_type) :: A
    ! sparse matrix descriptor
      type(psb_desc_type)   :: desc_A
    ! preconditioner
      type(mld_dprec_type)  :: P
    ! right-hand side and solution vectors
      real(kind(1.d0))      :: b(:), x(:)
    ... ...
    !
    ! initialize the parallel environment
      call psb_init(ictxt)
      call psb_info(ictxt,iam,np)
    ... ...
    !
    ! read and assemble the matrix A and the right-hand side b
    ! using PSBLAS routines for sparse matrix / vector management
    ... ...
    !
    ! initialize the default multi-level preconditioner, i.e. hybrid
    ! Schwarz, using RAS (with overlap 1 and ILU(0) on the blocks)
    ! as post-smoother and 4 block-Jacobi sweeps (with UMFPACK LU
    ! on the blocks) as distributed coarse-level solver
      call mld_precinit(P,'ML',info)
    !
    ! build the preconditioner
      call mld_precbld(A,desc_A,P,info)
    !
    ! set the solver parameters and the initial guess
      ... ...
    !
    ! solve Ax=b with preconditioned BiCGSTAB
      call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
      ... ...
    !
    ! deallocate the preconditioner
      call mld_precfree(P,info)
    !
    ! deallocate other data structures
      ... ...
    !
    ! exit the parallel environment
      call psb_exit(ictxt)
      stop

Different versions of multi-level preconditioners can be obtained by changing the default values of the preconditioner parameters. The code reported in Figure 3 shows how to set a three-level hybrid Schwarz preconditioner, which uses block Jacobi with ILU(0) on the local blocks as post-smoother, has a coarsest matrix replicated on the processors, and solves the coarsest-level system with the LU factorization from UMFPACK [9]. The number of levels is specified by using `mld_precinit`; the other preconditioner parameters are set by calling `mld_precset`. Note that the type of multilevel framework (i.e. multiplicative among the levels with post-smoothing only) is not specified since it is the default set by `mld_precinit`.

Figure 4 shows how to set a three-level additive Schwarz preconditioner, which uses RAS, with overlap 1 and ILU(0) on the blocks, as pre- and post-smoother, and applies five block-Jacobi sweeps, with the UMFPACK LU factorization on the blocks, as distributed coarsest-level solver. Again, `mld_precset` is used only to set non-default values of the parameters (see Tables 2-5). In both cases, the construction and the application of the preconditioner are carried out as for the default multi-level preconditioner. The code fragments shown in in Figures 3-4 are included in the example program file `mld_dexample_ml.f90` too.

Finally, Figure 5 shows the setup of a one-level additive Schwarz preconditioner, i.e. RAS with overlap 2. The corresponding example program is available in `mld_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 `examples/pdegen`.

Figure 3: Setup of a hybrid three-level Schwarz preconditioner.
    ... ...
    ! set a three-level hybrid Schwarz preconditioner, which uses
    ! block Jacobi (with ILU(0) on the blocks) as post-smoother,
    ! a coarsest matrix replicated on the processors, and the
    ! LU factorization from UMFPACK as coarse-level solver
      call mld_precinit(P,'ML',info,nlev=3)
      call_mld_precset(P,mld_smoother_type_,'BJAC',info)
      call mld_precset(P,mld_coarse_mat_,'REPL',info)
      call mld_precset(P,mld_coarse_solve_,'UMF',info)
    ... ...

Figure 4: Setup of an additive three-level Schwarz preconditioner.
    ... ...
    ! set a three-level additive Schwarz preconditioner, which uses
    ! RAS (with overlap 1 and ILU(0) on the blocks) as pre- and
    ! post-smoother, and 5 block-Jacobi sweeps (with UMFPACK LU
    ! on the blocks) as distributed coarsest-level solver
      call mld_precinit(P,'ML',info,nlev=3)
      call mld_precset(P,mld_ml_type_,'ADD',info)
      call_mld_precset(P,mld_smoother_pos_,'TWOSIDE',info)
      call mld_precset(P,mld_coarse_sweeps_,5,info)
    ... ...

Figure 5: Setup of a one-level Schwarz preconditioner.
    ... ...
    ! set RAS with overlap 2 and ILU(0) on the local blocks
      call mld_precinit(P,'AS',info)
      call mld_precset(P,mld_sub_ovr_,2,info)
    ... ...

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