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209 lines
9.9 KiB
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HREF="node2.html">Contents</A>
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HREF="node2.html">Contents</A></B>
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<H1><A NAME="SECTION00030000000000000000"></A><A NAME="sec:overview"></A>
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<BR>
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General Overview
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</H1><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The M<SMALL>ULTI</SMALL>L<SMALL>EVEL </SMALL>D<SMALL>OMAIN </SMALL>D<SMALL>ECOMPOSITION </SMALL>P<SMALL>ARALLEL </SMALL>P<SMALL>RECONDITIONERS </SMALL>P<SMALL>ACKAGE BASED ON
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</SMALL>PSBLAS (MLD2P4) provides parallel Algebraic MultiGrid (AMG) and Domain
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Decomposition preconditioners (see, e.g., [<A
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HREF="node36.html#Briggs2000">3</A>,<A
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HREF="node36.html#Stuben_01">24</A>,<A
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HREF="node36.html#dd2_96">22</A>]),
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to be used in the iterative solution of linear systems,
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</BIG></BIG></BIG>
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<BR>
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<DIV ALIGN="RIGHT" CLASS="mathdisplay">
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<!-- MATH
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\begin{equation}
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Ax=b,
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\end{equation}
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-->
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<A NAME="system1"></A>
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="system1"></A><IMG
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WIDTH="57" HEIGHT="30" BORDER="0"
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SRC="img2.png"
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ALT="\begin{displaymath}
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Ax=b,
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">1</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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where <SPAN CLASS="MATH"><IMG
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WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$"></SPAN> is a square, real or complex, sparse matrix.
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The name of the package comes from its original implementation, containing
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multilevel additive and hybrid Schwarz preconditioners, as well as one-level additive
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Schwarz preconditioners. The current version extends the original plan by including
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multilevel cycles and smoothers widely used in multigrid methods.
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The multilevel preconditioners implemented in MLD2P4 are obtained by combining
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AMG cycles with smoothers and coarsest-level solvers. The V-, W-, and
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K-cycles [<A
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HREF="node36.html#Briggs2000">3</A>,<A
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HREF="node36.html#Notay2008">20</A>] are available, which allow to define
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almost all the preconditioners in the package, including the multilevel hybrid
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Schwarz ones; a specific cycle is implemented to obtain multilevel additive
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Schwarz preconditioners. The Jacobi, hybrid
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forward/backward Gauss-Seidel, block-Jacobi, and additive Schwarz methods
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are available as smoothers. An algebraic approach is used to generate a hierarchy of
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coarse-level matrices and operators, without explicitly using any information on the
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geometry of the original problem, e.g., the discretization of a PDE. To this end,
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the smoothed aggregation technique [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<A
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HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>]
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is applied. Either exact or approximate solvers can be used on the coarsest-level
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system. Specifically, different sparse LU factorizations from external
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packages, and native incomplete LU factorizations and Jacobi, hybrid Gauss-Seidel,
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and block-Jacobi solvers are available. All smoothers can be also exploited as one-level
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preconditioners.
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">MLD2P4 is written in Fortran 2003, following an
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object-oriented design through the exploitation of features
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such as abstract data type creation, type extension, functional overloading, and
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dynamic memory management. The parallel implementation is based on a Single Program Multiple Data
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(SPMD) paradigm. Single and
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double precision implementations of MLD2P4 are available for both the
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real and the complex case, which can be used through a single
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interface.
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">MLD2P4 has been designed to implement scalable and easy-to-use
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multilevel preconditioners in the context of the PSBLAS (Parallel Sparse BLAS)
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computational framework [<A
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HREF="node36.html#psblas_00">15</A>,<A
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HREF="node36.html#PSBLAS3">14</A>]. PSBLAS provides basic linear algebra
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operators and data management facilities for distributed sparse matrices,
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as well as parallel Krylov solvers which can be used with the MLD2P4 preconditioners.
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The choice of PSBLAS has been mainly motivated by the need of having
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a portable and efficient software infrastructure implementing “de facto” standard
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parallel sparse linear algebra kernels, to pursue goals such as performance,
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portability, modularity ed extensibility in the development of the preconditioner
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package. On the other hand, the implementation of MLD2P4 has led to some
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revisions and extentions of the original PSBLAS kernels.
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The inter-process comunication required by MLD2P4 is encapsulated
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in the PSBLAS routines;
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therefore, MLD2P4 can be run on any parallel machine where PSBLAS
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implementations are available.
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">MLD2P4 has a layered and modular software architecture where three main layers can be
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identified. The lower layer consists of the PSBLAS kernels, the middle one implements
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the construction and application phases of the preconditioners, and the upper one
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provides a uniform interface to all the preconditioners.
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This architecture allows for different levels of use of the package:
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few black-box routines at the upper layer allow all users to easily
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build and apply any preconditioner available in MLD2P4;
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facilities are also available allowing expert users to extend the set of smoothers
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and solvers for building new versions of the preconditioners (see
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Section <A HREF="node33.html#sec:adding">7</A>).
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">We note that the user interface of MLD2P4 2.1 has been extended with respect to the
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previous versions in order to separate the construction of the multilevel hierarchy from
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the construction of the smoothers and solvers, and to allow for more flexibility
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at each level. The software architecture described in [<A
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HREF="node36.html#MLD2P4_TOMS">8</A>] has significantly
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evolved too, in order to fully exploit the Fortran 2003 features implemented in PSBLAS 3.
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However, compatibility with previous versions has been preserved.
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">This guide is organized as follows. General information on the distribution of the source
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code is reported in Section <A HREF="node4.html#sec:distribution">2</A>, while details on the configuration
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and installation of the package are given in Section <A HREF="node6.html#sec:building">3</A>. A short description
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of the preconditioners implemented in MLD2P4 is provided in Section <A HREF="node12.html#sec:background">4</A>,
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to help the users in choosing among them. The basics for building and applying the
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preconditioners with the Krylov solvers implemented in PSBLAS are reported
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in Section <A HREF="node16.html#sec:started">5</A>, where the Fortran codes of a few sample programs
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are also shown. A reference guide for the user interface routines is provided
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in Section <A HREF="node18.html#sec:userinterface">6</A>. Information on the extension of the package
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through the addition of new smoothers and solvers is reported in Section <A HREF="node33.html#sec:adding">7</A>.
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The error handling mechanism used by the package
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is briefly described in Section <A HREF="node34.html#sec:errors">8</A>. The copyright terms concerning the
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distribution and modification of MLD2P4 are reported in Appendix <A HREF="node35.html#sec:license">A</A>.
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</BIG></BIG></BIG>
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<P>
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