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335 lines
14 KiB
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335 lines
14 KiB
TeX
\section{Getting Started\label{sec:started}}
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\markboth{\textsc{MLD2P4 User's and Reference Guide}}
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{\textsc{\ref{sec:started} Getting Started}}
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We describe the basics for building and applying MLD2P4 one-level and multilevel
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(i.e., AMG) preconditioners with the Krylov solvers included in PSBLAS \cite{PSBLASGUIDE}.
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The following steps are required:
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\begin{enumerate}
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\item \emph{Declare the preconditioner data structure}. It is a derived data type,
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\verb|mld_|\-\emph{x}\verb|prec_| \verb|type|, where \emph{x} may be \verb|s|, \verb|d|, \verb|c|
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or \verb|z|, according to the basic data type of the sparse matrix
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(\verb|s| = real single precision; \verb|d| = real double precision;
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\verb|c| = complex single precision; \verb|z| = complex double precision).
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This data structure is accessed by the user only through the MLD2P4 routines,
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following an object-oriented approach.
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\item \emph{Allocate and initialize the preconditioner data structure, according to
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a preconditioner type chosen by the user}. This is performed by the routine
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\verb|init|, which also sets defaults for each preconditioner
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type selected by the user. The preconditioner types and the defaults associated
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with them are given in Table~\ref{tab:precinit}, where the strings used by
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\verb|init| to identify the preconditioner types are also given.
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Note that these strings are valid also if uppercase letters are substituted by
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corresponding lowercase ones.
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%\item \emph{Modify the aggregation parameters (for multilevel preconditioners only).}
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% This is performed by the routine \verb|mld_precset|.
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% This routine must be called only if the user wants to modify the default values
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% of the parameters associated with the aggregation hierarchy construction.
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% Examples of use of \verb|mld_precset| are given in
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% Section~\ref{sec:examples}; a complete list of all the
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% preconditioner parameters and their allowed and default values is provided in
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% Section~\ref{sec:userinterface}, Tables~\ref{tab:p_cycle}-\ref{tab:p_smoother_1}.
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\item \emph{Modify the selected preconditioner type, by properly setting
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preconditioner parameters.} This is performed by the routine \verb|set|.
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This routine must be called only if the user wants to modify the default values
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of the parameters associated with the selected preconditioner type, to obtain a variant
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of that preconditioner. Examples of use of \verb|set| are given in
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Section~\ref{sec:examples}; a complete list of all the
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preconditioner parameters and their allowed and default values is provided in
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Section~\ref{sec:userinterface}, Tables~\ref{tab:p_cycle}-\ref{tab:p_smoother_1}.
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\item \emph{Build the preconditioner for a given matrix}. If the selected preconditioner
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is multilevel, then two steps must be performed, as specified next.
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\begin{enumerate}
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\item[4.1] \emph{Build the aggregation hierarchy for a given matrix.} This is
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performed by the routine \verb|hierarchy_build|.
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\item[4.2] \emph{Build the preconditioner for a given matrix.} This is performed
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by the routine \verb|smoothers_build|.
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\end{enumerate}
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If the selected preconditioner is one-level, it is built in a single step,
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performed by the routine \verb|bld|.
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\item \emph{Apply the preconditioner at each iteration of a Krylov solver.}
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This is performed by the method \verb|apply|. When using the PSBLAS Krylov solvers,
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this step is completely transparent to the user, since \verb|apply| is called
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by the PSBLAS routine implementing the Krylov solver (\verb|psb_krylov|).
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\item \emph{Free the preconditioner data structure}. This is performed by
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the routine \verb|free|. This step is complementary to step 1 and should
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be performed when the preconditioner is no more used.
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\end{enumerate}
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All the previous routines are available as methods of the preconditioner object.
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A detailed description of them is given in Section~\ref{sec:userinterface}.
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Examples showing the basic use of MLD2P4 are reported in Section~\ref{sec:examples}.
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\begin{table}[h!]
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\begin{center}
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%{\small
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\begin{tabular}{|l|p{1.8cm}|p{8.2cm}|}
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\hline
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\textsc{type} & \textsc{string} & \textsc{default preconditioner} \\ \hline
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No preconditioner &\verb|'NONE'|& Considered to use the PSBLAS
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Krylov solvers with no preconditioner. \\ \hline
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Diagonal & \verb|'DIAG'| or \verb|'JACOBI'| & Diagonal preconditioner.
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For any zero diagonal entry of the matrix to be preconditioned,
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the corresponding entry of the preconditioner is set to~1.\\ \hline
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Gauss-Seidel & \verb|'GS'| & Hybrid Gauss-Seidel (forward), that is,
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global block Jacobi with
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Gauss-Seidel as local solver.\\ \hline
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Symmetrized Gauss-Seidel & \verb|'FBGS'| & Symmetrized hybrid Gauss-Seidel,that is,
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forward Gauss-Seidel followed by
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backward Gauss-Seidel.\\ \hline
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Block Jacobi & \verb|'BJAC'| & Block-Jacobi with ILU(0) on the local blocks.\\ \hline
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Additive Schwarz & \verb|'AS'| & Additive Schwarz (AS),
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with overlap~1 and ILU(0) on the local blocks. \\ \hline
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Multilevel &\verb|'ML'| & V-cycle with one hybrid forward Gauss-Seidel
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(GS) sweep as pre-smoother and one hybrid backward
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GS sweep as post-smoother, basic smoothed aggregation
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as coarsening algorithm, and LU (plus triangular solve)
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as coarsest-level solver. See the default values in
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Tables~\ref{tab:p_cycle}-\ref{tab:p_smoother_1}
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for further details of the preconditioner. \\
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\hline
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\end{tabular}
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%}
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\caption{Preconditioner types, corresponding strings and default choices.
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\label{tab:precinit}}
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\end{center}
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\end{table}
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Note that the module \verb|mld_prec_mod|, containing the definition of the
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preconditioner data type and the interfaces to the routines of MLD2P4,
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must be used in any program calling such routines.
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The modules \verb|psb_base_mod|, for the sparse matrix and communication descriptor
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data types, and \verb|psb_krylov_mod|, for interfacing with the
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Krylov solvers, must be also used (see Section~\ref{sec:examples}). \\
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\textbf{Remark 1.} Coarsest-level solvers based on the LU factorization,
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such as those implemented in UMFPACK, MUMPS, SuperLU, and SuperLU\_Dist,
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usually lead to smaller numbers of preconditioned Krylov
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iterations than inexact solvers, when the linear system comes from
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a standard discretization of basic scalar elliptic PDE problems. However,
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this does not necessarily correspond to the smallest execution time
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on parallel computers.
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% \ \\
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% \textbf{Remark 2.} The include path for MLD2P4 must override
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% those for PSBLAS, i.e.\ the former must come first in the sequence
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% passed to the compiler, as the MLD2P4 version of the Krylov solver
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% interfaces must override that of PSBLAS. This will change in the future
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% when the support for the \verb|class| statement becomes widespread in Fortran
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% compilers.
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\subsection{Examples\label{sec:examples}}
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The code reported in Figure~\ref{fig:ex1} shows how to set and apply the default
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multilevel preconditioner available in the real double precision version
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of MLD2P4 (see Table~\ref{tab:precinit}). This preconditioner is chosen
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by simply specifying \verb|'ML'| as the second argument of \verb|P%init|
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(a call to \verb|P%set| is not needed) and is applied with the CG
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solver provided by PSBLAS (the matrix of the system to be solved is
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assumed to be positive definite). As previously observed, the modules
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\verb|psb_base_mod|, \verb|mld_prec_mod| and \verb|psb_krylov_mod|
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must be used by the example program.
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The part of the code concerning the
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reading and assembling of the sparse matrix and the right-hand side vector, performed
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through the PSBLAS routines for sparse matrix and vector management, is not reported
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here for brevity; the statements concerning the deallocation of the PSBLAS
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data structure are neglected too.
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The complete code can be found in the example program file \verb|mld_dexample_ml.f90|,
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in the directory \verb|examples/fileread| of the MLD2P4 implementation (see
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Section~\ref{sec:ex_and_test}). A sample test problem along with the relevant
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input data is available in \verb|examples/fileread/runs|.
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For details on the use of the PSBLAS routines, see the PSBLAS User's
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Guide~\cite{PSBLASGUIDE}.
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The setup and application of the default multilevel preconditioner
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for the real single precision and the complex, single and double
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precision, versions are obtained with straightforward modifications of the previous
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example (see Section~\ref{sec:userinterface} for details). If these versions are installed,
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the corresponding codes are available in \verb|examples/fileread/|.
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\begin{figure}[tbp]
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\begin{center}
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\begin{minipage}{.90\textwidth}
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{\small
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\begin{verbatim}
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use psb_base_mod
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use mld_prec_mod
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use psb_krylov_mod
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... ...
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!
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! sparse matrix
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type(psb_dspmat_type) :: A
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! sparse matrix descriptor
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type(psb_desc_type) :: desc_A
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! preconditioner
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type(mld_dprec_type) :: P
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! right-hand side and solution vectors
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type(psb_d_vect_type) :: b, x
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... ...
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!
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! initialize the parallel environment
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call psb_init(ictxt)
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call psb_info(ictxt,iam,np)
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... ...
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!
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! read and assemble the spd matrix A and the right-hand side b
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! using PSBLAS routines for sparse matrix / vector management
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... ...
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!
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! initialize the default multilevel preconditioner, i.e. V-cycle
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! with basic smoothed aggregation, 1 hybrid forward/backward
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! GS sweep as pre/post-smoother and UMFPACK as coarsest-level
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! solver
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call P%init('ML',info)
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!
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! build the preconditioner
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call P%hierarchy_build(A,desc_A,info)
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call P%smoothers_build(A,desc_A,info)
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!
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! set the solver parameters and the initial guess
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... ...
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!
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! solve Ax=b with preconditioned CG
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call psb_krylov('CG',A,P,b,x,tol,desc_A,info)
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... ...
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!
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! deallocate the preconditioner
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call P%free(info)
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!
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! deallocate other data structures
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... ...
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!
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! exit the parallel environment
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call psb_exit(ictxt)
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stop
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\end{verbatim}
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}
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\end{minipage}
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\caption{setup and application of the default multilevel preconditioner (example 1).
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\label{fig:ex1}}
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\end{center}
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\end{figure}
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Different versions of the multilevel preconditioner can be obtained by changing
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the default values of the preconditioner parameters. The code reported in
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Figure~\ref{fig:ex2} shows how to set a V-cycle preconditioner
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which applies 1 block-Jacobi sweep as pre- and post-smoother,
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and solves the coarsest-level system with 8 block-Jacobi sweeps.
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Note that the ILU(0) factorization (plus triangular solve) is used as
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local solver for the block-Jacobi sweeps, since this is the default associated
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with block-Jacobi and set by~\verb|P%init|.
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Furthermore, specifying block-Jacobi as coarsest-level
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solver implies that the coarsest-level matrix is distributed
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among the processes.
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Figure~\ref{fig:ex3} shows how to set a W-cycle preconditioner which
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applies 2 hybrid Gauss-Seidel sweeps as pre- and post-smoother,
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and solves the coarsest-level system with the multifrontal LU factorization
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implemented in MUMPS. It is specified that the coarsest-level
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matrix is distributed, since MUMPS can be used on both
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replicated and distributed matrices, and by default
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it is used on replicated ones.
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%Note the use of the parameter \verb|pos|
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%to specify a property only for the pre-smoother or the post-smoother
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%(see Section~\ref{sec:precset} for more details).
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The code fragments shown in Figures~\ref{fig:ex2} and \ref{fig:ex3} are
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included in the example program file \verb|mld_dexample_ml.f90| too.
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Finally, Figure~\ref{fig:ex4} shows the setup of a one-level
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additive Schwarz preconditioner, i.e., RAS with overlap 2.
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Note also that a Krylov method different from CG must be used to solve
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the preconditioned system, since the preconditione in nonsymmetric.
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The corresponding example program is available in the file
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\verb|mld_dexample_1lev.f90|.
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For all the previous preconditioners, example programs where the sparse matrix and
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the right-hand side are generated by discretizing a PDE with Dirichlet
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boundary conditions are also available in the directory \verb|examples/pdegen|.
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% \ \\
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% \textbf{Remark 2.} Any PSBLAS-based program using the basic preconditioners
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% implemented in PSBLAS 3.0, i.e.\ the diagonal and block-Jacobi ones,
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% can use the diagonal and block-Jacobi preconditioners
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% implemented in MLD2P4 without change in the code.
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% The PSBLAS-based program must be only recompiled
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% and linked to the MLD2P4 library.
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% \\
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\begin{figure}[tbh]
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\begin{center}
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\begin{minipage}{.90\textwidth}
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{\small
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\begin{verbatim}
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... ...
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! build a V-cycle preconditioner with 1 block-Jacobi sweep (with
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! ILU(0) on the blocks) as pre- and post-smoother, and 8 block-Jacobi
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! sweeps (with ILU(0) on the blocks) as coarsest-level solver
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call P%init('ML',info)
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call_P%set('SMOOTHER_TYPE','BJAC',info)
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call P%set('COARSE_SOLVE','BJAC',info)
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call P%set('COARSE_SWEEPS',8,info)
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call P%hierarchy_build(A,desc_A,info)
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call P%smoothers_build(A,desc_A,info)
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... ...
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\end{verbatim}
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}
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\end{minipage}
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\caption{setup of a multilevel preconditioner\label{fig:ex2}}
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\end{center}
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\end{figure}
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\begin{figure}[h!]
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\begin{center}
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\begin{minipage}{.90\textwidth}
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{\small
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\begin{verbatim}
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... ...
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! build a W-cycle preconditioner with 2 hybrid Gauss-Seidel sweeps
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! as pre- and post-smoother, a distributed coarsest
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! matrix, and MUMPS as coarsest-level solver
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call P%init('ML',info)
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call P%set('ML_CYCLE','WCYCLE',info)
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call P%set('SMOOTHER_TYPE','FBGS',info)
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call P%set('SMOOTHER_SWEEPS',2,info)
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call P%set('COARSE_SOLVE','MUMPS',info)
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call P%set('COARSE_MAT','DIST',info)
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call P%hierarchy_build(A,desc_A,info)
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call P%smoothers_build(A,desc_A,info)
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... ...
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\end{verbatim}
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}
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\end{minipage}
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\caption{setup of a multilevel preconditioner\label{fig:ex3}}
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\end{center}
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\end{figure}
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\begin{figure}[h!]
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\begin{center}
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\begin{minipage}{.90\textwidth}
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{\small
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\begin{verbatim}
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... ...
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! set RAS with overlap 2 and ILU(0) on the local blocks
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call P%init('AS',info)
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call P%set('SUB_OVR',2,info)
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call P%bld(A,desc_A,info)
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... ...
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! solve Ax=b with preconditioned BiCGSTAB
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call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info)
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\end{verbatim}
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}
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\end{minipage}
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\caption{setup of a one-level Schwarz preconditioner.\label{fig:ex4}}
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\end{center}
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\end{figure}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "userguide"
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%%% End:
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