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233 lines
7.8 KiB
TeX
233 lines
7.8 KiB
TeX
\section{Preconditioner routines}
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\label{sec:precs}
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% \section{Preconditioners}
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\label{sec:psprecs}
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PSBLAS contains the implementation of many preconditioning
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techniques some of which are very flexible thanks to the presence of
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many parameters that is possible to adjust to fit the user's needs:
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\begin{itemize}
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\item Diagonal Scaling
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\item Block Jacobi with ILU(0) factorization
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\item Additive Schwarz with the Restricted Additive Schwarz and
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Additive Schwarz with Harmonic extensions;
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\item Two-Level Additive Schwarz; this is actually a family of
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preconditioners since there is the possibility to choose between
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many variants.
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\end{itemize}
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\subroutine{psb\_precset}{Sets the preconditioner type}
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\syntax{call psb\_precset}{prec, ptype, iv, rs, ierr}
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\begin{description}
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\item[\bf On Entry]
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\item[prec]
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a pronditioner data structure \precdata.
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\item[ptype] the type of preconditioner.
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: a character string, see usage notes.
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\item[iv] integer parameters for the precondtioner.
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: an integer array, see usage notes.
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\item[rs]
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Specified as: a long precision real number.
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\item[ierr]
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Scope: {\bf global} \\
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Type: {\bf required}\\
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\end{description}
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\section*{Usage Notes}
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The PSBLAS 2.0 contains a number of preconditioners, ranging from a
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simple diagonal scaling to 2-level domain decomposition. These
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preconditioners may use the SuperLU or the UMFPACK software, if
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installed; see~\cite{SUPERLU,UMFPACK}.
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Legal inputs to this subroutine are interpreted depending on the
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$ptype$ string as follows\footnote{The string is case-insensitive}:
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\begin{description}
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\item[NONE] No preconditioning, i.e. the preconditioner is just a copy
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operator.
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\item[DIAG] Diagonal scaling; each entry of the input vector is
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multiplied by the reciprocal of the sum of the absolute values of
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the coefficients in the corresponding row of matrix $A$;
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\item[ILU] Precondition by the incomplete LU factorization of the
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block-diagonal of matrix $A$, where block boundaries are determined
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by the data allocation boundaries for each process; requires no
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communication. Only $ILU(0)$ is currently implemented.
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\item[AS] Additive Schwarz preconditioner (see~\cite{PARA04}); in this
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case the user may specify additional flags through the integer
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vector \verb|ir| as follows:
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\begin{description}
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\item[$iv(1)$] Number of overlap levels, an integer $novr>=0$, default
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$novr=1$.
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\item[$iv(2)$] Restriction operator, legal values: \verb|psb_halo_|,
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\verb|psb_none_|; default: \verb|psb_halo_|
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\item[$iv(3)$] Prolongation operator, legal values: \verb|psb_none_|,
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\verb|psb_sum_|, \verb|psb_avg_|, default: \verb|psb_none_|
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\item[$iv(4)$] Factorization type, legal values: \verb|f_ilu_n_|,
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\verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|.
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\end{description}
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Note that the default corresponds to a Restricted Additive Schwarz
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preconditioner with $ILU(0)$ and 1 level of overlap.
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\item[2L] Second level of a multilevel preconditioner, see below
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\end{description}
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If a multilevel preconditioner is desired, the user should call
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\verb|psb_precset| twice, the first time choosing an AS variant, and
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a second time specifying
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$ptype=2L$ with the following optional parameters in $iv$ (see
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also~\cite{APNUM06,DD2}):
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\begin{description}
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\item[$iv(1)$] Type of multilevel correction, legal values: \verb|no_ml_|,
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\verb|add_ml_prec_|, \verb|mult_ml_prec_|,
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default: \verb|mult_ml_prec_|;
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\item[$iv(2)$] Aggregation algorithm, legal values: \verb|loc_aggr_|;
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\item[$iv(3)$] Smoother type, legal values: \verb|no_smth_|,
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\verb|smth_omg_|, default: \verb|smth_omg_|;
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\item[$iv(4)$] Coarse matrix allocation, legal values:
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\verb|mat_distr_|, \verb|mat_repl_|, default: \verb|mat_distr_|
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\item[$iv(5)$] Smoother position, legal values: \verb|pre_smooth_|,
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\verb|post_smooth_|, \verb|smooth_both_|, default:
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\verb|post_smooth_|
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\item[$iv(6)$] Factorization type (for coarse matrix), legal values: \verb|f_ilu_n_|,
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\verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|;
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\item[$iv(7)$] Number of Jacobi sweeps for coarse system correction,
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default 1.
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\item[$rs$] Set the smoother parameter $\omega$ a user defined value;
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default: esitimate with the infinity norm of matrix $A$.
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\end{description}
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The 2-level preconditioners are based on the idea of building a
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coarse-space approximation $A_C$ of the matrix $A$; given a set $W_C$
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of coarse vertices, with size $n_C$, and a suitable restriction
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operator $R_C \in \Re^{n_C \times n}$, $A_C$ is defined as
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\[
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A_C=R_C A R_C^T .
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\]
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The prolongator $R_C^T$ is built with the smoothed aggregation technique,
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in which we start from a tentative prolongator that simply maps
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fine-level entries onto their aggregates $P_C$; if the user chooses
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\verb|no_smth_| this is the prolongator used, otherwise it is
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multiplied by a smoother \[ S = I - \omega D^{-1} A \], where $D$ is
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the diagonal of $A$ and $\omega$ may be imposed by the user or
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estimated internally.
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The coarse space correction may be added to the fine level solution
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\verb|add_ml_prec_|
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\[
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M_{2L-A}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
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\]
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or it can be composed in a multiplicative framework as a pre-smoothed
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correction
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\[
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M_{2L-H1}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1},
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\]
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post-smoothed correction
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\[
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M_{2L-H2}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
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\]
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or two-sided.
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\subroutine{psb\_precbld}{Builds a preconditioner}
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\syntax{call psb\_precbld}{a, desc\_a, prec, info, upd}
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\begin{description}
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\item[\bf On Entry]
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\item[a] the system sparse matrix.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a sparse matrix data structure \spdata.
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\item[desc\_a] the problem communication descriptor.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a communication descriptor data structure \descdata.
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\item[upd]
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Scope: {\bf global} \\
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Type: {\bf optional}\\
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Specified as: a character.
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\end{description}
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\begin{description}
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\item[\bf On Return]
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\item[prec] the preconditioner.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a precondtioner data structure \precdata\\
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\item[info] the return error code.\\
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Scope: {\bf global} \\
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Type: {\bf required}\\
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Specified as: an integer, upon successful completion $info=0$ \\
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\end{description}
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\subroutine{psb\_precaply}{Preconditioner application routine}
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\syntax{call psb\_precaply}{prec,x,y,desc\_a,info,trans,work}
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\syntax{call psb\_precaply}{prec,x,desc\_a,info,trans}
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\begin{description}
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\item[\bf On Entry]
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\item[prec] the preconditioner.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a preconditioner data structure \precdata.
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\item[x] the source vector.
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Scope: {\bf local} \\
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Type: {\bf require}\\
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Specified as: a double precision array.
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\item[desc\_a] the problem communication descriptor.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a communication data structure \descdata.
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\item[trans]
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Scope: {\bf } \\
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Type: {\bf optional}\\
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Specified as: a character.
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\item[work] an optional work space
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Scope: {\bf local} \\
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Type: {\bf optional}\\
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Specified as: a double precision array.
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\end{description}
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\begin{description}
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\item[\bf On Return]
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\item[y] the destination vector.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a double precision array.
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\item[info] the return error code.\\
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: an integer, upon successful completion $info=0$ .\\
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\end{description}
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\subroutine{psb\_prec\_descr}{Prints a description of current preconditioner}
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\syntax{call psb\_prec\_descr}{prec}
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\begin{description}
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\item[\bf On Entry]
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\item[prec] the preconditioner.
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Scope: {\bf local} \\
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Type: {\bf required}\\
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Specified as: a preconditioner data structure \precdata.
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\end{description}
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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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