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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
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<TITLE>Examples</TITLE>
<META NAME="description" CONTENT="Examples">
<TITLE>Smoothers and coarsest-level solvers</TITLE>
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@@ -19,296 +19,216 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00071000000000000000"></A><A NAME="sec:examples"></A>
<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
<BR>
Examples
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The code reported in Figure&nbsp;<A HREF="#fig:ex1">2</A> shows how to set and apply the default
multi-level preconditioner available in the real double precision version
of MLD2P4 (see Table&nbsp;<A HREF="#tab:precinit">1</A>). This preconditioner is chosen
by simply specifying <code>'ML'</code> as the second argument of <code>P%init</code>
(a call to <code>P%set</code> 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
<code>psb_base_mod</code>, <code>mld_prec_mod</code> and <code>psb_krylov_mod</code>
must be used by the example program.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">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 <code>mld_dexample_ml.f90</code>,
in the directory <code>examples/fileread</code> of the MLD2P4 implementation (see
Section&nbsp;<A HREF="node10.html#sec:ex_and_test">3.5</A>). A sample test problem along with the relevant
input data is available in <code>examples/fileread/runs</code>.
For details on the use of the PSBLAS routines, see the PSBLAS User's
Guide&nbsp;[<A
HREF="node27.html#PSBLASGUIDE">13</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The setup and application of the default multi-level 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&nbsp;<A HREF="node15.html#sec:userinterface">6</A> for details). If these versions are installed,
the corresponding codes are available in <code>examples/fileread/</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex1"></A><A NAME="535"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
setup and application of the default multi-level preconditioner (example 1).
</CAPTION>
<TR><TD>
Smoothers and coarsest-level solvers
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,␍a hybrid version of the forward and backward Gauss-Seidel methods, and the␍additive Schwarz (AS) ones (see, e.g., [<A
HREF="node29.html#Saad_book">20</A>,<A
HREF="node29.html#dd2_96">21</A>]). ␍␍The hybrid Gauss-Seidel␍version is considered because the original Gauss-Seidel method is inherently sequential.␍At each iteration of the hybrid version, each parallel process uses the most recent values␍of its own local variables and the values of the non-local variables computed at the␍previous iteration, obtained by exchanging data with other processes before␍the beginning of the current iteration.␍␍In the AS methods, the index space <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"> is divided into <IMG
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img50.png"
ALT="$m_k$">␍subsets <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$"> of size <IMG
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$n_{k,i}$">, possibly␍overlapping. For each <IMG
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"> we consider the restriction␍operator <!-- MATH
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
-->
<IMG
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img53.png"
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">␍that maps a vector <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$"> to the vector <IMG
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img55.png"
ALT="$x_i^k$"> made of the components of <IMG
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img54.png"
ALT="$x^k$">␍with indices in <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">, and the prolongation operator␍<!-- MATH
$P^k_i = (R_i^k)^T$
-->
<IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img56.png"
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build␍<!-- MATH
$A_i^k=R_i^kA^kP_i^k$
-->
<IMG
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"> to the index␍space <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img51.png"
ALT="$\Omega^k_i$">.␍The classical AS preconditioner <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> is defined as␍</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
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
type(psb_d_vect_type) :: b, x
... ...
!
! initialize the parallel environment
call psb_init(ictxt)
call psb_info(ictxt,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 multi-level 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('ML',info)
!
! build the preconditioner
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
<!-- MATH
\begin{displaymath}
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
\end{displaymath}
-->
!
! set the solver parameters and the initial guess
... ...
!
! solve Ax=b with preconditioned CG
call psb_krylov('CG',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(ictxt)
stop
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
<IMG
WIDTH="219" HEIGHT="59" BORDER="0"
SRC="img59.png"
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Different versions of the multi-level preconditioner can be obtained by changing
the default values of the preconditioner parameters. The code reported in
Figure&nbsp;<A HREF="#fig:ex2">3</A> 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&nbsp;<code>P%init</code>.
Furthermore, specifying block-Jacobi as coarsest-level
solver implies that the coarsest-level matrix is distributed
among the processes.
Figure&nbsp;<A HREF="#fig:ex3">4</A> shows how to set a W-cycle preconditioner which
applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother,
and solves the coarsest-level system with the multifrontal LU factorization
implemented in MUMPS. It is specified that the coarsest-level
matrix is distributed, since MUMPS can be used on both
replicated and distributed matrices, and by default
it is used on replicated ones. The code fragments shown in Figures&nbsp;<A HREF="#fig:ex2">3</A> and <A HREF="#fig:ex3">4</A> are
included in the example program file <code>mld_dexample_ml.f90</code> too.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Finally, Figure&nbsp;<A HREF="#fig:ex4">5</A> 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
<code>mld_dexample_1lev.f90</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">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 <code>examples/pdegen</code>.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex2"></A><A NAME="537"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! 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('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)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV>
<P>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex3"></A><A NAME="539"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
setup of a multi-level preconditioner</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! 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('ML',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','MUMPS',info)
call P%set('COARSE_MAT','DIST',info)
call P%hierarchy_build(A,desc_A,info)
call P%smoothers_build(A,desc_A,info)
... ...
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:ex4"></A><A NAME="541"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
setup of a one-level Schwarz preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
</DIV><TABLE WIDTH="90%">
<TR><TD>
<PRE>
... ...
! set RAS with overlap 2 and ILU(0) on the local blocks
call P%init('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)
</PRE>
</TD></TR>
</TABLE>
<DIV ALIGN="CENTER">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate␍inverse of <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$"> is usually considered instead of <IMG
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img61.png"
ALT="$(A_i^k)^{-1}$">.␍The setup of <IMG
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img58.png"
ALT="$M^k_{AS}$"> during the multilevel build phase␍involves␍</FONT></FONT></FONT>
<UL>
<LI>the definition of the index subspaces <IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img62.png"
ALT="$\Omega_i^k$"> and of the corresponding ␍ operators <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img63.png"
ALT="$R_i^k$"> (and <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img64.png"
ALT="$P_i^k$">);
</LI>
<LI>the computation of the submatrices <IMG
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img60.png"
ALT="$A_i^k$">;
</LI>
<LI>the computation of their inverses (usually approximated␍ through some form of incomplete factorization).
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍The computation of <!-- MATH
$z^k=M^k_{AS}w^k$
-->
<IMG
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img65.png"
ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
$w^k \in \mathbb{R}^{n_k}$
-->
<IMG
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img66.png"
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the␍multilevel application phase, requires␍</FONT></FONT></FONT>
<UL>
<LI>the restriction of <IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img67.png"
ALT="$w^k$"> to the subspaces <!-- MATH
$\mathbb{R}^{n_{k,i}}$
-->
<IMG
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img68.png"
ALT="$\mathbb{R}^{n_{k,i}}$">,␍ i.e. <!-- MATH
$w_i^k = R_i^{k} w^k$
-->
<IMG
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img69.png"
ALT="$w_i^k = R_i^{k} w^k$">;
</LI>
<LI>the computation of the vectors <!-- MATH
$z_i^k=(A_i^k)^{-1} w_i^k$
-->
<IMG
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img70.png"
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
</LI>
<LI>the prolongation and the sum of the previous vectors,␍ i.e. <!-- MATH
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
-->
<IMG
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img71.png"
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
</LI>
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍Variants of the classical AS method, which use modifications of the␍restriction and prolongation operators, are also implemented in MLD2P4.␍Among them, the Restricted AS (RAS) preconditioner usually␍outperforms the classical AS preconditioner in terms of convergence␍rate and of computation and communication time on parallel distributed-memory␍computers, and is therefore the most widely used among the AS␍preconditioners&nbsp;[<A
HREF="node29.html#CAI_SARKIS">6</A>]. ␍␍Direct solvers based on sparse LU factorizations, implemented in the␍third-party libraries reported in Section&nbsp;<A HREF="node7.html#sec:third-party">3.2</A>, can be applied␍as coarsest-level solvers by MLD2P4. Native inexact solvers based on␍incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,␍and block Jacobi preconditioners are also available. Direct solvers usually␍lead to more effective preconditioners in terms of algorithmic scalability;␍however, this does not guarantee parallel efficiency.␍
</FONT></FONT></FONT><HR>
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