|
|
|
|
@ -11,10 +11,14 @@ many parameters that is possible to adjust to fit the user's needs:
|
|
|
|
|
\item Block Jacobi with ILU(0) factorization
|
|
|
|
|
\item Additive Schwarz with the Restricted Additive Schwarz and
|
|
|
|
|
Additive Schwarz with Harmonic extensions;
|
|
|
|
|
\item Two-Level Additive Schwarz; this is actually a family of
|
|
|
|
|
preconditioners since there is the possibility to choose between
|
|
|
|
|
many variants.
|
|
|
|
|
\end{itemize}
|
|
|
|
|
The PSBLAS library is incorporating a package of two-level Additive
|
|
|
|
|
Schwarz preconditioners called MD2P4; this is actually a family of
|
|
|
|
|
preconditioners since there is the possibility to choose between
|
|
|
|
|
many variants, and is currently in an experimental state. Its
|
|
|
|
|
documentation is planned to appear after stabilization of the
|
|
|
|
|
package, which will characterize release 2.1 of our library.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
@ -77,61 +81,61 @@ $ptype$ string as follows\footnote{The string is case-insensitive}:
|
|
|
|
|
\end{description}
|
|
|
|
|
Note that the default corresponds to a Restricted Additive Schwarz
|
|
|
|
|
preconditioner with $ILU(0)$ and 1 level of overlap.
|
|
|
|
|
\item[2L] Second level of a multilevel preconditioner, see below
|
|
|
|
|
\end{description}
|
|
|
|
|
If a multilevel preconditioner is desired, the user should call
|
|
|
|
|
\verb|psb_precset| twice, the first time choosing an AS variant, and
|
|
|
|
|
a second time specifying
|
|
|
|
|
$ptype=2L$ with the following optional parameters in $iv$ (see
|
|
|
|
|
also~\cite{APNUM06,DD2}):
|
|
|
|
|
\begin{description}
|
|
|
|
|
\item[$iv(1)$] Type of multilevel correction, legal values: \verb|no_ml_|,
|
|
|
|
|
\verb|add_ml_prec_|, \verb|mult_ml_prec_|,
|
|
|
|
|
default: \verb|mult_ml_prec_|;
|
|
|
|
|
\item[$iv(2)$] Aggregation algorithm, legal values: \verb|loc_aggr_|;
|
|
|
|
|
\item[$iv(3)$] Smoother type, legal values: \verb|no_smth_|,
|
|
|
|
|
\verb|smth_omg_|, default: \verb|smth_omg_|;
|
|
|
|
|
\item[$iv(4)$] Coarse matrix allocation, legal values:
|
|
|
|
|
\verb|mat_distr_|, \verb|mat_repl_|, default: \verb|mat_distr_|
|
|
|
|
|
\item[$iv(5)$] Smoother position, legal values: \verb|pre_smooth_|,
|
|
|
|
|
\verb|post_smooth_|, \verb|smooth_both_|, default:
|
|
|
|
|
\verb|post_smooth_|
|
|
|
|
|
\item[$iv(6)$] Factorization type (for coarse matrix), legal values: \verb|f_ilu_n_|,
|
|
|
|
|
\verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|;
|
|
|
|
|
\item[$iv(7)$] Number of Jacobi sweeps for coarse system correction,
|
|
|
|
|
default 1.
|
|
|
|
|
\item[$rs$] Set the smoother parameter $\omega$ a user defined value;
|
|
|
|
|
default: esitimate with the infinity norm of matrix $A$.
|
|
|
|
|
%% \item[2L] Second level of a multilevel preconditioner, see below
|
|
|
|
|
%% \end{description}
|
|
|
|
|
%% If a multilevel preconditioner is desired, the user should call
|
|
|
|
|
%% \verb|psb_precset| twice, the first time choosing an AS variant, and
|
|
|
|
|
%% a second time specifying
|
|
|
|
|
%% $ptype=2L$ with the following optional parameters in $iv$ (see
|
|
|
|
|
%% also~\cite{APNUM06,DD2}):
|
|
|
|
|
%% \begin{description}
|
|
|
|
|
%% \item[$iv(1)$] Type of multilevel correction, legal values: \verb|no_ml_|,
|
|
|
|
|
%% \verb|add_ml_prec_|, \verb|mult_ml_prec_|,
|
|
|
|
|
%% default: \verb|mult_ml_prec_|;
|
|
|
|
|
%% \item[$iv(2)$] Aggregation algorithm, legal values: \verb|loc_aggr_|;
|
|
|
|
|
%% \item[$iv(3)$] Smoother type, legal values: \verb|no_smth_|,
|
|
|
|
|
%% \verb|smth_omg_|, default: \verb|smth_omg_|;
|
|
|
|
|
%% \item[$iv(4)$] Coarse matrix allocation, legal values:
|
|
|
|
|
%% \verb|mat_distr_|, \verb|mat_repl_|, default: \verb|mat_distr_|
|
|
|
|
|
%% \item[$iv(5)$] Smoother position, legal values: \verb|pre_smooth_|,
|
|
|
|
|
%% \verb|post_smooth_|, \verb|smooth_both_|, default:
|
|
|
|
|
%% \verb|post_smooth_|
|
|
|
|
|
%% \item[$iv(6)$] Factorization type (for coarse matrix), legal values: \verb|f_ilu_n_|,
|
|
|
|
|
%% \verb|f_slu_|, \verb|f_umf_|, default: \verb|f_ilu_n_|;
|
|
|
|
|
%% \item[$iv(7)$] Number of Jacobi sweeps for coarse system correction,
|
|
|
|
|
%% default 1.
|
|
|
|
|
%% \item[$rs$] Set the smoother parameter $\omega$ a user defined value;
|
|
|
|
|
%% default: esitimate with the infinity norm of matrix $A$.
|
|
|
|
|
\end{description}
|
|
|
|
|
The 2-level preconditioners are based on the idea of building a
|
|
|
|
|
coarse-space approximation $A_C$ of the matrix $A$; given a set $W_C$
|
|
|
|
|
of coarse vertices, with size $n_C$, and a suitable restriction
|
|
|
|
|
operator $R_C \in \Re^{n_C \times n}$, $A_C$ is defined as
|
|
|
|
|
\[
|
|
|
|
|
A_C=R_C A R_C^T .
|
|
|
|
|
\]
|
|
|
|
|
The prolongator $R_C^T$ is built with the smoothed aggregation technique,
|
|
|
|
|
in which we start from a tentative prolongator that simply maps
|
|
|
|
|
fine-level entries onto their aggregates $P_C$; if the user chooses
|
|
|
|
|
\verb|no_smth_| this is the prolongator used, otherwise it is
|
|
|
|
|
multiplied by a smoother \[ S = I - \omega D^{-1} A \], where $D$ is
|
|
|
|
|
the diagonal of $A$ and $\omega$ may be imposed by the user or
|
|
|
|
|
estimated internally.
|
|
|
|
|
The coarse space correction may be added to the fine level solution
|
|
|
|
|
\verb|add_ml_prec_|
|
|
|
|
|
\[
|
|
|
|
|
M_{2L-A}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
|
|
|
|
\]
|
|
|
|
|
or it can be composed in a multiplicative framework
|
|
|
|
|
(\verb|mult_ml_prec_|)as a pre-smoothed correction (\verb|pre_smooth_|)
|
|
|
|
|
\[
|
|
|
|
|
M_{2L-H1}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1},
|
|
|
|
|
\]
|
|
|
|
|
post-smoothed correction (\verb|post_smooth_|)
|
|
|
|
|
\[
|
|
|
|
|
M_{2L-H2}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
|
|
|
|
\]
|
|
|
|
|
or two-sided for symmetric matrices (\verb|smooth_both_|).
|
|
|
|
|
%% The 2-level preconditioners are based on the idea of building a
|
|
|
|
|
%% coarse-space approximation $A_C$ of the matrix $A$; given a set $W_C$
|
|
|
|
|
%% of coarse vertices, with size $n_C$, and a suitable restriction
|
|
|
|
|
%% operator $R_C \in \Re^{n_C \times n}$, $A_C$ is defined as
|
|
|
|
|
%% \[
|
|
|
|
|
%% A_C=R_C A R_C^T .
|
|
|
|
|
%% \]
|
|
|
|
|
%% The prolongator $R_C^T$ is built with the smoothed aggregation technique,
|
|
|
|
|
%% in which we start from a tentative prolongator that simply maps
|
|
|
|
|
%% fine-level entries onto their aggregates $P_C$; if the user chooses
|
|
|
|
|
%% \verb|no_smth_| this is the prolongator used, otherwise it is
|
|
|
|
|
%% multiplied by a smoother \[ S = I - \omega D^{-1} A \], where $D$ is
|
|
|
|
|
%% the diagonal of $A$ and $\omega$ may be imposed by the user or
|
|
|
|
|
%% estimated internally.
|
|
|
|
|
%% The coarse space correction may be added to the fine level solution
|
|
|
|
|
%% \verb|add_ml_prec_|
|
|
|
|
|
%% \[
|
|
|
|
|
%% M_{2L-A}^{-1} = M_{C}^{-1} + M_{1L}^{-1}.
|
|
|
|
|
%% \]
|
|
|
|
|
%% or it can be composed in a multiplicative framework
|
|
|
|
|
%% (\verb|mult_ml_prec_|)as a pre-smoothed correction (\verb|pre_smooth_|)
|
|
|
|
|
%% \[
|
|
|
|
|
%% M_{2L-H1}^{-1} = M_{C}^{-1} + \left( I - M_{C}^{-1}A \right) M_{1L}^{-1},
|
|
|
|
|
%% \]
|
|
|
|
|
%% post-smoothed correction (\verb|post_smooth_|)
|
|
|
|
|
%% \[
|
|
|
|
|
%% M_{2L-H2}^{-1} = M_{1L}^{-1} + \left( I - M_{1L}^{-1}A \right) M_{C}^{-1}.
|
|
|
|
|
%% \]
|
|
|
|
|
%% or two-sided for symmetric matrices (\verb|smooth_both_|).
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Salvatore Filippone
19 years ago