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309 lines
13 KiB
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309 lines
13 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 multi-level
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Schwarz 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|mld_precinit|, which also sets defaults for each preconditioner
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type selected by the user. The defaults associated to each preconditioner
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type are given in Table~\ref{tab:precinit}, where the strings used by
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\verb|mld_precinit| 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 selected preconditioner type, by properly setting
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preconditioner parameters.} 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 to the selected preconditioner type, to obtain a variant
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of the preconditioner. 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_type}-\ref{tab:p_coarse}.
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\item \emph{Build the preconditioner for a given matrix.} This is performed by
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the routine \verb|mld_precbld|.
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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 routine \verb|mld_precaply|. When using the PSBLAS Krylov solvers,
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this step is completely transparent to the user, since \verb|mld_precaply| 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|mld_| \verb|precfree|. 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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A detailed description of the above routines 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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Note that the Fortran 95 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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\ \\
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\textbf{Remark 1.} The coarsest-level solver used by the default two-level
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preconditioner has been chosen by taking into account that, on parallel
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machines, it often leads to the smallest execution time when applied to
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linear systems coming from finite-difference discretizations of basic
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elliptic PDE problems, considered as standard tests for multi-level Schwarz
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preconditioners \cite{aaecc_07,apnum_07}. However, this solver does
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not necessarily correspond to the smallest number of iterations of the
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preconditioned Krylov method, which is usually obtained by applying
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a direct solver to the coarsest-level system, e.g.\ based on the LU
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factorization (see Section~\ref{sec:userinterface}
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for the coarsest-level solvers available in MLD2P4).
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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, e.g.\ the latter 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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\begin{table}[th]
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\begin{center}
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%{\small
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\begin{tabular}{|l|l|p{7.8cm}|}
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\hline
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\textsc{type} & \textsc{string} & \textsc{default preconditioner} \\ \hline
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No preconditioner &\verb|'NOPREC'|& Considered only to use the PSBLAS
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Krylov solvers with no preconditioner. \\ \hline
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Diagonal & \verb|'DIAG'| & --- \\ \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'| & Restricted Additive Schwarz (RAS),
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with overlap 1 and ILU(0) on the local blocks. \\ \hline
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Multilevel &\verb|'ML'| & Multi-level hybrid preconditioner (additive on the
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same level and multiplicative through the levels),
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with post-smoothing only.
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Number of levels: 2.
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Post-smoother: RAS with overlap 1 and ILU(0)
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on the local blocks.
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Aggregation: decoupled smoothed aggregation with
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threshold $\theta = 0$.
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Coarsest matrix: distributed among the processors.
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Coarsest-level solver:
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4 sweeps of the block-Jacobi solver,
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with LU (or ILU) factorization of the blocks
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(UMFPACK for the double precision versions and
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SuperLU for the single precision ones, if the packages
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have been installed; ILU(0), otherwise). \\
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\hline
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\end{tabular}
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%}
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\end{center}
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\caption{Preconditioner types, corresponding strings and default choices.
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\label{tab:precinit}}
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\end{table}
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\subsection{Examples\label{sec:examples}}
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The code reported in Figure~\ref{fig:ex_default} shows how to set and apply the default
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multi-level 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 second argument of \verb|mld_precinit|
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(a call to \verb|mld_precset| is not needed) and is applied with the BiCGSTAB
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solver provided by PSBLAS. As previously observed, the modules \verb|psb_base_mod|,
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\verb|mld_prec_mod| and \verb|psb_krylov_mod| 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 tree (see
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Section~\ref{sec:ex_and_test}).
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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 multi-level
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preconditioners 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 Fortran 95 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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real(kind(1.d0)) :: 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 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 multi-level preconditioner, i.e. hybrid
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! Schwarz, using RAS (with overlap 1 and ILU(0) on the blocks)
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! as post-smoother and 4 block-Jacobi sweeps (with UMFPACK LU
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! on the blocks) as distributed coarse-level solver
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call mld_precinit(P,'ML',info)
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!
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! build the preconditioner
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call mld_precbld(A,desc_A,P,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 BiCGSTAB
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call psb_krylov('BICGSTAB',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 mld_precfree(P,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 multi-level Schwarz preconditioner.
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\label{fig:ex_default}}
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\end{center}
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\end{figure}
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Different versions of multi-level preconditioners 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:ex_3lh} shows how to set a three-level hybrid Schwarz
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preconditioner, which uses block Jacobi with ILU(0) on the
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local blocks as post-smoother, has a coarsest matrix replicated on the processors,
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and solves the coarsest-level system with the LU factorization from UMFPACK~\cite{UMFPACK}.
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The number of levels is specified by using \verb|mld_precinit|; the other
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preconditioner parameters are set by calling \verb|mld_precset|. Note that
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the type of multilevel framework (i.e.\ multiplicative among the levels
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with post-smoothing only) is not specified since it is the default
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set by \verb|mld_precinit|.
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Figure~\ref{fig:ex_3la} shows how to
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set a three-level additive Schwarz preconditioner,
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which uses RAS, with overlap 1 and ILU(0) on the blocks,
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as pre- and post-smoother, and applies five block-Jacobi sweeps, with
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the UMFPACK LU factorization on the blocks, as distributed coarsest-level
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solver. Again, \verb|mld_precset| is used only to set
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non-default values of the parameters (see Tables~\ref{tab:p_type}-\ref{tab:p_coarse}).
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In both cases, the construction and the application of the preconditioner
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are carried out as for the default multi-level preconditioner.
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The code fragments shown in in Figures~\ref{fig:ex_3lh}-\ref{fig:ex_3la} 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:ex_1l} shows the setup of a one-level
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additive Schwarz preconditioner, i.e.\ RAS with overlap 2. The corresponding
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example program is available in \verb|mld_dexample_| \verb|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 3.} Any PSBLAS-based program using the basic preconditioners
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implemented in PSBLAS 2.0, 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 any 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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! set a three-level hybrid Schwarz preconditioner, which uses
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! block Jacobi (with ILU(0) on the blocks) as post-smoother,
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! a coarsest matrix replicated on the processors, and the
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! LU factorization from UMFPACK as coarse-level solver
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call mld_precinit(P,'ML',info,nlev=3)
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call_mld_precset(P,mld_smoother_type_,'BJAC',info)
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call mld_precset(P,mld_coarse_mat_,'REPL',info)
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call mld_precset(P,mld_coarse_solve_,'UMF',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 hybrid three-level Schwarz preconditioner.\label{fig:ex_3lh}}
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\end{center}
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\end{figure}
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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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! set a three-level additive Schwarz preconditioner, which uses
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! RAS (with overlap 1 and ILU(0) on the blocks) as pre- and
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! post-smoother, and 5 block-Jacobi sweeps (with UMFPACK LU
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! on the blocks) as distributed coarsest-level solver
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call mld_precinit(P,'ML',info,nlev=3)
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call mld_precset(P,mld_ml_type_,'ADD',info)
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call_mld_precset(P,mld_smoother_pos_,'TWOSIDE',info)
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call mld_precset(P,mld_coarse_sweeps_,5,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 an additive three-level Schwarz preconditioner.\label{fig:ex_3la}}
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\end{center}
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\end{figure}
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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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! set RAS with overlap 2 and ILU(0) on the local blocks
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call mld_precinit(P,'AS',info)
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call mld_precset(P,mld_sub_ovr_,2,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 one-level Schwarz preconditioner.\label{fig:ex_1l}}
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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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