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Modified description of preconditioners, to defer 2-level to upcoming
release 2.1.
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Salvatore Filippone
+61
-57
@@ -11,10 +11,14 @@ many parameters that is possible to adjust to fit the user's needs:
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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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The PSBLAS library is incorporating a package of two-level Additive
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Schwarz preconditioners called MD2P4; 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, and is currently in an experimental state. Its
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documentation is planned to appear after stabilization of the
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package, which will characterize release 2.1 of our library.
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@@ -77,61 +81,61 @@ $ptype$ string as follows\footnote{The string is case-insensitive}:
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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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%% \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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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
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(\verb|mult_ml_prec_|)as a pre-smoothed correction (\verb|pre_smooth_|)
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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 (\verb|post_smooth_|)
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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 for symmetric matrices (\verb|smooth_both_|).
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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
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%% (\verb|mult_ml_prec_|)as a pre-smoothed correction (\verb|pre_smooth_|)
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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 (\verb|post_smooth_|)
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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 for symmetric matrices (\verb|smooth_both_|).
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+1850
-2167
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