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\section{Getting Started\label{sec: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{}.
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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_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 a default preconditioner for each preconditioner
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type selected by the user. The default preconditioner associated to each preconditioner
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type is listed in Table~\ref{tab:precinit}; the string used by \verb|mld_precinit|
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to identify each preconditioner type is also given.
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\item \emph{Choose a specific preconditioner within the selected preconditioner type, by setting
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the 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 default preconditioner.
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An example of use of \verb|mld_precset| is given in
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Section~\ref{sec:examples}, Figure~\ref{fig:example2}; 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:highlevel}.
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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{Deallocate the preconditioner data structure}. This is performed by
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the routine \verb|mld_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:highlevel}.
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Note that the Fortran 95 module \verb|mld_prec_mod| must be used in the program
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calling the MLD2P4 routines. Furthermore, to apply MLD2P4 with the Krylov solvers
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from PSBLAS, the module \verb|psb_krylov_mod| must be used too.
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Examples showing the basic use of MLD2P4 are reported in Section~\ref{sec:examples}.
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\begin{table}[th]
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{
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\begin{center}
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\begin{tabular}{|l|l|p{6.7cm}|}
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\hline
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Type & String & 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.) \\
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Diagonal & \verb|'DIAG'| & --- \\
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Block Jacobi & \verb|'BJAC'| & Block Jacobi with ILU(0) on the local blocks.\\
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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. \\
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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. Number of levels: 2;
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post-smoother: block-Jacobi preconditioner with ILU(0)
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on the local blocks; coarsest matrix: distributed among the
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processors; corase-level solver: 4 sweeps of the block-Jacobi
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solver, with the UMFPACK LU factorization on the blocks. \\
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\hline
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\end{tabular}
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\end{center}
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}
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\caption{Preconditioner types and default choices.\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:example1} shows how to set and apply the MLD2P4 default
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multi-level preconditioner, i.e.\ the two-level hybrid post-smoothed Schwarz preconditioner,
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having block-Jacobi with ILU(0) on the blocks as basic preconditioner,
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a coarse matrix distributed among the processors, and four block-Jacobi
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sweeps, with the UMFPACK sparse LU factorization on the blocks, as approximate coarse-level solver.
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The choice of this preconditioner is made
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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).
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The preconditioner is applied with the BiCGSTAB solver provided by PSBLAS.
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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 data structure
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are neglected too.
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The complete code can be found in the example program file \verb|example_ml_default.f90|
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in the directory \textbf{XXXXXX (SPECIFICARE).} Note that the modules \verb|psb_base_mod|
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and \verb|psb_util_mod| at the beginning of the code are required by PSBLAS.
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For details on the use of the PSBLAS routines, see the PSBLAS User's Guide \cite{}.
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\begin{figure}[tbp]
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\begin{center}
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{\small
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\begin{verbatim}
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use psb_base_mod
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use psb_util_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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... ...
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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
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! side b using PSBLAS routines for sparse matrix /
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! vector management
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... ...
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!
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! initialize the default multi-level preconditioner
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! (two-level hybrid Schwarz, with ILU(0) as post-smoother
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! and 4 Block-Jacobi sweeps, with ILU(0) on the blocks,
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! as distributed coarsest-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 psb_precbld(A,P,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 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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\caption{Setup and application of the default multilevel Schwarz preconditioner.
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\label{fig:example1}}
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\end{center}
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\end{figure}
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Different versions of 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:example2} shows how to set a three-level hybrid Schwarz preconditioner
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using RAS with overlap 1 as post-smoother, a coarsest matrix replicated
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on the processors and the LU factorization from UMFPACK as coarse-level solver.
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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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which corresponds to the hybrid multilevel preconditioner); the type
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of one-level AS preconditioner used as smoother (i.e.\ RAS)
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and its ``position'' (i.e.\ pre-smoother) are not specified since they
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are chosen by default when \verb|mld_precinit| is called.
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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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As a further example, we report in Figure~\ref{fig:example3} the code
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concerning the setup of a three-level additive multi-level preconditioner,
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using ILU(0) as pre- and post-smoother, a distributed coarsest matrix and
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five block-Jacobi sweeps as coarsest-level solver, with
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ILU(0) on the local blocks. Again, \verb|mld_precset| is used only to set
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the values of the parameters that are not default values.
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For a detailed description of the parameters associated to a preconditioner
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type, including their allowed and default values, the user is referred to
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\textbf{SPECIFICARE.}
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An example program including the code fragments
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shown in in Figures~\ref{fig:example2} and \ref{fig:example3} is in
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\verb|XXX/.../example_3lev.f90|. \textbf{COMPLETARE. Fare un programma solo
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per i due esempi, in cui uno e' commentato e l'altro no.}
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One more example program, showing the setup and application of a one-level
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additive Schwarz preconditioner can be found in \verb|XXX/.../example_1lev.f90|.
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\textbf{COMPLETARE}.
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\begin{figure}[tbp]
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\begin{center}
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{\small
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\begin{verbatim}
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... ...
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! setup a three-level hybrid Schwarz preconditioner,
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! using RAS with overlap 1 as post-smoother, a coarsest
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! matrix replicated on the processors, and the LU
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! 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_smooth_type_,'AS',info)
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call mld_precset(P,mld_n_ovr_,1,info)
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call mld_precset(P,mld_coarse_mat,'REPL')
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call mld_precset(P,mld_coarse_solve,'UMF')
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... ...
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\end{verbatim}
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}
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\caption{Setup of a hybrid three-level Schwarz preconditioner.\label{fig:example2}}
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\end{center}
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\end{figure}
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\begin{figure}[tbp]
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\begin{center}
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{\small
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\begin{verbatim}
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... ...
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! setup a three-level additive Schwarz preconditioner,
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! using ILU(0) as pre- and post-smoother, five block-Jacobi
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! sweeps as distributed coarsest-level solver, with ILU(0)
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! on the local blocks
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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_smooth_pos_,'TWOSIDE',info)
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call mld_precset(P,mld_n_ovr_,1,info)
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call mld_precset(P,mld_coarse_sweeps,5)
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call mld_precset(P,mld_coarse_subsolve,'UMF')
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... ...
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\end{verbatim}
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}
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\caption{Setup of an additive three-level Schwarz preconditioner.\label{fig:example3}}
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\end{center}
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\end{figure}
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\ \\
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\textbf{Note.} Any PSBLAS-based program using the basic preconditioners implemented in PSBLAS 2.0,
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i.e.\ the diagonal and block-Jacobi ones, can use the diagonal and block-Jacobi preconditioners
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implemented in MLD2P4 without any change in the code. The PSBLAS-base program must e only recompiled
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and linked to the MLD2P4 library.
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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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