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Fix from Mac coding to Unix coding.
This commit is contained in:
Salvatore Filippone
+63
-5
@@ -54,17 +54,75 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
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<BR>
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Multigrid Background
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</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍␍Multigrid preconditioners, coupled with Krylov iterative␍solvers, are widely used in the parallel solution of large and sparse linear systems,␍because of their optimality in the solution of linear systems arising from the␍discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.␍Optimality, also known as algorithmic scalability, is the property ␍of having a computational cost per iteration that depends linearly on␍the problem size, and a convergence rate that is independent of the problem size.␍␍Multigrid preconditioners are based on a recursive application of a two-grid process␍consisting of smoother iterations and a coarse-space (or coarse-level) correction.␍The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,␍or more complex subspace-correction methods, such as the Schwarz ones.␍The coarse-space correction consists of solving, in an appropriately chosen␍coarse space, the residual equation associated with the approximate solution computed␍by the smoother, and of using the solution of this equation to correct the␍previous approximation. The transfer of information between the original␍(fine) space and the coarse one is performed by using suitable restriction and␍prolongation operators. The construction of the coarse space and the corresponding␍transfer operators is carried out by applying a so-called coarsening algorithm to the system␍matrix. Two main approaches can be used to perform coarsening: the geometric approach,␍which exploits the knowledge of some physical grid associated with the matrix␍and requires the user to define transfer operators from the fine␍to the coarse level and vice versa, and the algebraic approach, which builds␍the coarse-space correction and the associate transfer operators using only matrix␍information. The first approach may be difficult when the system comes from␍discretizations on complex geometries;␍furthermore, ad hoc one-level smoothers may be required to get an efficient␍interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients␍are considered. The second approach performs a fully automatic coarsening and enforces the␍interplay between fine and coarse level by suitably choosing the coarse space and␍the coarse-to-fine interpolation (see, e.g., [<A
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</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners, coupled with Krylov iterative
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solvers, are widely used in the parallel solution of large and sparse linear systems,
|
||||
because of their optimality in the solution of linear systems arising from the
|
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discretization of scalar elliptic Partial Differential Equations (PDEs) on regular grids.
|
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Optimality, also known as algorithmic scalability, is the property
|
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of having a computational cost per iteration that depends linearly on
|
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the problem size, and a convergence rate that is independent of the problem size.
|
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Multigrid preconditioners are based on a recursive application of a two-grid process
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consisting of smoother iterations and a coarse-space (or coarse-level) correction.
|
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The smoothers may be either basic iterative methods, such as the Jacobi and Gauss-Seidel ones,
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or more complex subspace-correction methods, such as the Schwarz ones.
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The coarse-space correction consists of solving, in an appropriately chosen
|
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coarse space, the residual equation associated with the approximate solution computed
|
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by the smoother, and of using the solution of this equation to correct the
|
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previous approximation. The transfer of information between the original
|
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(fine) space and the coarse one is performed by using suitable restriction and
|
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prolongation operators. The construction of the coarse space and the corresponding
|
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transfer operators is carried out by applying a so-called coarsening algorithm to the system
|
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matrix. Two main approaches can be used to perform coarsening: the geometric approach,
|
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which exploits the knowledge of some physical grid associated with the matrix
|
||||
and requires the user to define transfer operators from the fine
|
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to the coarse level and vice versa, and the algebraic approach, which builds
|
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the coarse-space correction and the associate transfer operators using only matrix
|
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information. The first approach may be difficult when the system comes from
|
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discretizations on complex geometries;
|
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furthermore, ad hoc one-level smoothers may be required to get an efficient
|
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interplay between fine and coarse levels, e.g., when matrices with highly varying coefficients
|
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are considered. The second approach performs a fully automatic coarsening and enforces the
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interplay between fine and coarse level by suitably choosing the coarse space and
|
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the coarse-to-fine interpolation (see, e.g., [<A
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HREF="node29.html#Briggs2000">3</A>,<A
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HREF="node29.html#Stuben_01">23</A>,<A
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HREF="node29.html#dd2_96">21</A>] for details.)␍␍MLD2P4 uses a pure algebraic approach, based on the smoothed ␍aggregation algorithm [<A
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HREF="node29.html#dd2_96">21</A>] for details.)
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">MLD2P4 uses a pure algebraic approach, based on the smoothed
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aggregation algorithm [<A
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HREF="node29.html#BREZINA_VANEK">2</A>,<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>],␍for building the sequence of coarse matrices and transfer operators,␍starting from the original one.␍A decoupled version of this algorithm is implemented, where the smoothed␍aggregation is applied locally to each submatrix [<A
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HREF="node29.html#TUMINARO_TONG">24</A>].␍A brief description of the AMG preconditioners implemented in MLD2P4 is given in ␍Sections <A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="node14.html#sec:smoothers">4.3</A>. For further details the reader␍is referred to [<A
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HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>],
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for building the sequence of coarse matrices and transfer operators,
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starting from the original one.
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A decoupled version of this algorithm is implemented, where the smoothed
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aggregation is applied locally to each submatrix [<A
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HREF="node29.html#TUMINARO_TONG">24</A>].
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A brief description of the AMG preconditioners implemented in MLD2P4 is given in
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Sections <A HREF="node12.html#sec:multilevel">4.1</A>-<A HREF="node14.html#sec:smoothers">4.3</A>. For further details the reader
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is referred to [<A
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HREF="node29.html#para_04">4</A>,<A
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HREF="node29.html#aaecc_07">5</A>,<A
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HREF="node29.html#apnum_07">7</A>,<A
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HREF="node29.html#MLD2P4_TOMS">8</A>].␍␍We note that optimal multigrid preconditioners do not necessarily correspond␍to minimum execution times in a parallel setting. Indeed, to obtain effective parallel␍multigrid preconditioners, a tradeoff between the optimality and the cost of building and␍applying the smoothers and the coarse-space corrections must be achieved. Effective␍parallel preconditioners require algorithmic scalability to be coupled with implementation␍scalability, i.e., a computational cost per iteration which remains (almost) constant as␍the number of parallel processors increases.␍␍␍</FONT></FONT></FONT>
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HREF="node29.html#MLD2P4_TOMS">8</A>].
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">We note that optimal multigrid preconditioners do not necessarily correspond
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to minimum execution times in a parallel setting. Indeed, to obtain effective parallel
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multigrid preconditioners, a tradeoff between the optimality and the cost of building and
|
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applying the smoothers and the coarse-space corrections must be achieved. Effective
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parallel preconditioners require algorithmic scalability to be coupled with implementation
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scalability, i.e., a computational cost per iteration which remains (almost) constant as
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the number of parallel processors increases.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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+98
-29
@@ -54,7 +54,11 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<BR>
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AMG preconditioners
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</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍In order to describe the AMG preconditioners available in MLD2P4, we consider a␍linear system␍</FONT></FONT></FONT>
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</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
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linear system
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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@@ -73,30 +77,41 @@ Ax=b,
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<TD WIDTH=10 ALIGN="RIGHT">
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(2)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <!-- MATH
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<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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where <!-- MATH
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$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
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-->
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<IMG
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;␍for ease of presentation we assume <IMG
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ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
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for ease of presentation we assume <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$"> is real, but the␍results are valid for the complex case as well. ␍␍Let us assume as finest index space the set of row (column) indices of <IMG
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ALT="$A$"> is real, but the
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results are valid for the complex case as well.
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</FONT></FONT></FONT>
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<P>
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">, i.e.,␍<!-- MATH
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ALT="$A$">, i.e.,
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<!-- MATH
|
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$\Omega = \{1, 2, \ldots, n\}$
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-->
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<IMG
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WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$\Omega = \{1, 2, \ldots, n\}$">. ␍Any algebraic multilevel preconditioners implemented in MLD2P4 generates␍a hierarchy of index spaces and a corresponding hierarchy of matrices,␍</FONT></FONT></FONT>
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ALT="$\Omega = \{1, 2, \ldots, n\}$">.
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Any algebraic multilevel preconditioners implemented in MLD2P4 generates
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a hierarchy of index spaces and a corresponding hierarchy of matrices,
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</FONT></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
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\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
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\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
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\end{displaymath}
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-->
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@@ -106,13 +121,16 @@ Ax=b,
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ALT="\begin{displaymath}\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev}, \end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍by using the information contained in <IMG
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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by using the information contained in <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$">, without assuming any␍knowledge of the geometry of the problem from which <IMG
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ALT="$A$">, without assuming any
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knowledge of the geometry of the problem from which <IMG
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WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$A$"> originates.␍A vector space <!-- MATH
|
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ALT="$A$"> originates.
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A vector space <!-- MATH
|
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$\mathbb{R}^{n_{k}}$
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-->
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<IMG
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@@ -121,27 +139,32 @@ Ax=b,
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ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
|
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img9.png"
|
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ALT="$\Omega^k$">,␍where <IMG
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ALT="$\Omega^k$">,
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where <IMG
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WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="$n_k$"> is the size of <IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$">.␍For all <IMG
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ALT="$\Omega^k$">.
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For all <IMG
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WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$k < nlev$">, a restriction operator and a prolongation one are built,␍which connect two levels <IMG
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ALT="$k < nlev$">, a restriction operator and a prolongation one are built,
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which connect two levels <IMG
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WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.png"
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ALT="$k$"> and <IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="$k+1$">:␍</FONT></FONT></FONT>
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ALT="$k+1$">:
|
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</FONT></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER">
|
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<!-- MATH
|
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\begin{displaymath}
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P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad ␍ R^k \in \mathbb{R}^{n_{k+1}\times n_k};
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P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
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R^k \in \mathbb{R}^{n_{k+1}\times n_k};
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\end{displaymath}
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-->
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@@ -151,10 +174,13 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
R^k \in \mathbb{R}^{n_{k+1}\
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ALT="\begin{displaymath}␍ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad ␍ R^k \in \mathbb{R}^{n_{k+1}\times n_k};␍\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍the matrix <IMG
|
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
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the matrix <IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
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ALT="$A^{k+1}$"> is computed by using the previous operators according␍to the Galerkin approach, i.e.,␍</FONT></FONT></FONT>
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ALT="$A^{k+1}$"> is computed by using the previous operators according
|
||||
to the Galerkin approach, i.e.,
|
||||
</FONT></FONT></FONT>
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<BR><P></P>
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<DIV ALIGN="CENTER">
|
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<!-- MATH
|
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@@ -169,25 +195,34 @@ A^{k+1}=R^kA^kP^k.
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ALT="\begin{displaymath}␍ A^{k+1}=R^kA^kP^k.␍\end{displaymath}">
|
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</DIV>
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<BR CLEAR="ALL">
|
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<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍In the current implementation of MLD2P4 we have <IMG
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
In the current implementation of MLD2P4 we have <IMG
|
||||
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
ALT="$R^k=(P^k)^T$">␍A smoother with iteration matrix <IMG
|
||||
ALT="$R^k=(P^k)^T$">
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A smoother with iteration matrix <IMG
|
||||
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img18.png"
|
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ALT="$M^k$"> is set up at each level <IMG
|
||||
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$k < nlev$">, and a solver␍is set up at the coarsest level, so that they are ready for application ␍(for example, setting up a solver based on the <IMG
|
||||
ALT="$k < nlev$">, and a solver
|
||||
is set up at the coarsest level, so that they are ready for application
|
||||
(for example, setting up a solver based on the <IMG
|
||||
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img19.png"
|
||||
ALT="$LU$"> factorization means computing␍and storing the <IMG
|
||||
ALT="$LU$"> factorization means computing
|
||||
and storing the <IMG
|
||||
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$L$"> and <IMG
|
||||
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$U$"> factors). The construction of the hierarchy of AMG components␍described so far corresponds to the so-called build phase of the preconditioner.␍␍</FONT></FONT></FONT>
|
||||
ALT="$U$"> factors). The construction of the hierarchy of AMG components
|
||||
described so far corresponds to the so-called build phase of the preconditioner.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
@@ -195,26 +230,60 @@ Application phase of a V-cycle preconditioner.</CAPTION>
|
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<TR><TD>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
$\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad \=\quad \\[-3mm]␍procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]␍\>if $\left(k \ne nlev \right)$\ then \\[1mm]␍\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]␍\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]␍\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]␍\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]␍\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]␍\>else \\[1mm]␍\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]␍\>endif \\[1mm]␍\>return $u^k$\ \\[1mm]␍end␍\end{tabbing}␍\end{minipage}␍}$
|
||||
$\framebox{
|
||||
\begin{minipage}{.85\textwidth}
|
||||
\begin{tabbing}
|
||||
\quad \=\quad \=\quad \=\quad \\[-3mm]
|
||||
procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
|
||||
\>if $\left(k \ne nlev \right)$\ then \\[1mm]
|
||||
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
|
||||
\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
|
||||
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
|
||||
\>else \\[1mm]
|
||||
\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
|
||||
\>endif \\[1mm]
|
||||
\>return $u^k$\ \\[1mm]
|
||||
end
|
||||
\end{tabbing}
|
||||
\end{minipage}
|
||||
}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
|
||||
...mm]␍\>endif \\ [1mm]␍\>return $u^k$\ \\ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
|
||||
...mm]␍\>endif \ [1mm]␍\>return $u^k$ \ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
|
||||
|
||||
</DIV></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍The components produced in the build phase may be combined in several ways␍to obtain different multilevel preconditioners;␍this is done in the application phase, i.e., in the computation of a vector␍of type <IMG
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The components produced in the build phase may be combined in several ways
|
||||
to obtain different multilevel preconditioners;
|
||||
this is done in the application phase, i.e., in the computation of a vector
|
||||
of type <IMG
|
||||
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$w=B^{-1}v$">, where <IMG
|
||||
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$B$"> denotes the preconditioner, usually within an iteration␍of a Krylov solver [<A
|
||||
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as␍V-cycle, is given in Figure <A HREF="#fig:application_alg">1</A>. In this case, a single iteration␍of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,␍in the pre-smoothing and post-smoothing phases); however, different choices can be␍performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle␍and W-cycle [<A
|
||||
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described␍in [<A
|
||||
HREF="node29.html#Notay2008">19</A>]. ␍␍␍</FONT></FONT></FONT><HR>
|
||||
ALT="$B$"> denotes the preconditioner, usually within an iteration
|
||||
of a Krylov solver [<A
|
||||
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as
|
||||
V-cycle, is given in Figure <A HREF="#fig:application_alg">1</A>. In this case, a single iteration
|
||||
of the same smoother is used before and after the the recursive call to the V-cycle (i.e.,
|
||||
in the pre-smoothing and post-smoothing phases); however, different choices can be
|
||||
performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
|
||||
and W-cycle [<A
|
||||
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
|
||||
in [<A
|
||||
HREF="node29.html#Notay2008">19</A>].
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html229"
|
||||
HREF="node13.html">
|
||||
|
||||
+108
-40
@@ -54,25 +54,37 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
|
||||
<BR>
|
||||
Smoothed Aggregation
|
||||
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍In order to define the prolongator <IMG
|
||||
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$">, used to compute␍the coarse-level matrix <IMG
|
||||
ALT="$P^k$">, used to compute
|
||||
the coarse-level matrix <IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation␍algorithm described in [<A
|
||||
ALT="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
|
||||
algorithm described in [<A
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].␍The basic idea of this algorithm is to build a coarse set of indices␍<IMG
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>].
|
||||
The basic idea of this algorithm is to build a coarse set of indices
|
||||
<IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img26.png"
|
||||
ALT="$\Omega^{k+1}$"> by suitably grouping the indices of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$"> into disjoint␍subsets (aggregates), and to define the coarse-to-fine space transfer operator␍<IMG
|
||||
ALT="$\Omega^k$"> into disjoint
|
||||
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant␍prolongation operator, with the aim of improving the quality of the coarse-space correction.␍␍Three main steps can be identified in the smoothed aggregation procedure:␍</FONT></FONT></FONT>
|
||||
|
||||
ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
|
||||
prolongation operator, with the aim of improving the quality of the coarse-space correction.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Three main steps can be identified in the smoothed aggregation procedure:
|
||||
</FONT></FONT></FONT>
|
||||
<OL>
|
||||
<LI>aggregation of the indices of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
@@ -98,8 +110,12 @@ Smoothed Aggregation
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$">.
|
||||
</LI>
|
||||
</OL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍ ␍In order to perform the coarsening step, the smoothed aggregation algorithm␍described in [<A
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,␍each index <!-- MATH
|
||||
</OL><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to perform the coarsening step, the smoothed aggregation algorithm
|
||||
described in [<A
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
|
||||
each index <!-- MATH
|
||||
$j \in \Omega^{k+1}$
|
||||
-->
|
||||
<IMG
|
||||
@@ -111,22 +127,26 @@ Smoothed Aggregation
|
||||
ALT="$\Omega^k_j$"> of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$">,␍consisting of a suitably chosen index <!-- MATH
|
||||
ALT="$\Omega^k$">,
|
||||
consisting of a suitably chosen index <!-- MATH
|
||||
$i \in \Omega^k$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img29.png"
|
||||
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a␍strongly-coupled neighborood of <IMG
|
||||
ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
|
||||
strongly-coupled neighborood of <IMG
|
||||
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i$">, i.e.,␍</FONT></FONT></FONT>
|
||||
ALT="$i$">, i.e.,
|
||||
</FONT></FONT></FONT>
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
|
||||
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
||||
\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
@@ -139,35 +159,54 @@ Smoothed Aggregation
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(3)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍for a given threshold <!-- MATH
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
for a given threshold <!-- MATH
|
||||
$\theta \in [0,1]$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$\theta \in [0,1]$"> (see [<A
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).␍Since this algorithm has a sequential nature, a decoupled␍version of it is applied, where each processor independently executes␍the algorithm on the set of indices assigned to it in the initial data␍distribution. This version is embarrassingly parallel, since it does not require any data ␍communication. On the other hand, it may produce some nonuniform aggregates␍and is strongly dependent on the number of processors and on the initial partitioning␍of the matrix <IMG
|
||||
HREF="node29.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
|
||||
Since this algorithm has a sequential nature, a decoupled
|
||||
version of it is applied, where each processor independently executes
|
||||
the algorithm on the set of indices assigned to it in the initial data
|
||||
distribution. This version is embarrassingly parallel, since it does not require any data
|
||||
communication. On the other hand, it may produce some nonuniform aggregates
|
||||
and is strongly dependent on the number of processors and on the initial partitioning
|
||||
of the matrix <IMG
|
||||
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for␍MLD2P4, since it has been shown to produce good results in practice␍[<A
|
||||
ALT="$A$">. Nevertheless, this parallel algorithm has been chosen for
|
||||
MLD2P4, since it has been shown to produce good results in practice
|
||||
[<A
|
||||
HREF="node29.html#aaecc_07">5</A>,<A
|
||||
HREF="node29.html#apnum_07">7</A>,<A
|
||||
HREF="node29.html#TUMINARO_TONG">24</A>].␍␍The prolongator <IMG
|
||||
HREF="node29.html#TUMINARO_TONG">24</A>].
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The prolongator <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"> is built starting from a tentative prolongator␍<!-- MATH
|
||||
ALT="$P^k$"> is built starting from a tentative prolongator
|
||||
<!-- MATH
|
||||
$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img33.png"
|
||||
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as␍</FONT></FONT></FONT>
|
||||
ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
|
||||
</FONT></FONT></FONT>
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{ \begin{array}{ll}␍1 & \quad \mbox{if} \; i \in \Omega^k_j, \\␍0 & \quad \mbox{otherwise},␍\end{array} \right.
|
||||
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
||||
\left\{ \begin{array}{ll}
|
||||
1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
|
||||
0 & \quad \mbox{otherwise},
|
||||
\end{array} \right.
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
@@ -175,36 +214,41 @@ Smoothed Aggregation
|
||||
WIDTH="287" HEIGHT="51" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
|
||||
...ega^k_j, \\ ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
|
||||
...ega^k_j, \ ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
|
||||
\end{displaymath}"></TD>
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(4)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
where <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img28.png"
|
||||
ALT="$\Omega^k_j$"> is the aggregate of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$">␍corresponding to the index <!-- MATH
|
||||
ALT="$\Omega^k$">
|
||||
corresponding to the index <!-- MATH
|
||||
$j \in \Omega^{k+1}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img27.png"
|
||||
ALT="$j \in \Omega^{k+1}$">.␍<IMG
|
||||
ALT="$j \in \Omega^{k+1}$">.
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"> is obtained by applying to <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$\bar{P}^k$"> a smoother␍<!-- MATH
|
||||
ALT="$\bar{P}^k$"> a smoother
|
||||
<!-- MATH
|
||||
$S^k \in \mathbb{R}^{n_k \times n_k}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img36.png"
|
||||
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:␍</FONT></FONT></FONT>
|
||||
ALT="$S^k \in \mathbb{R}^{n_k \times n_k}$">:
|
||||
</FONT></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
@@ -219,12 +263,17 @@ P^k = S^k \bar{P}^k,
|
||||
ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍in order to remove nonsmooth components from the range of the prolongator,␍and hence to improve the convergence properties of the multi-level␍method [<A
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
in order to remove nonsmooth components from the range of the prolongator,
|
||||
and hence to improve the convergence properties of the multi-level
|
||||
method [<A
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node29.html#Stuben_01">23</A>].␍A simple choice for <IMG
|
||||
HREF="node29.html#Stuben_01">23</A>].
|
||||
A simple choice for <IMG
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$S^k$"> is the damped Jacobi smoother:␍</FONT></FONT></FONT>
|
||||
ALT="$S^k$"> is the damped Jacobi smoother:
|
||||
</FONT></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
@@ -239,25 +288,35 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
where <IMG
|
||||
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img40.png"
|
||||
ALT="$D^k$"> is the diagonal matrix with the same diagonal entries as <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$">,␍<!-- MATH
|
||||
ALT="$A^k$">,
|
||||
<!-- MATH
|
||||
$A^k_F = (\bar{a}_{ij}^k)$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img42.png"
|
||||
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as␍</FONT></FONT></FONT>
|
||||
ALT="$A^k_F = (\bar{a}_{ij}^k)$"> is the filtered matrix defined as
|
||||
</FONT></FONT></FONT>
|
||||
<BR>
|
||||
<DIV ALIGN="RIGHT">
|
||||
|
||||
<!-- MATH
|
||||
\begin{equation}
|
||||
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\␍ 0 & \mbox{otherwise},␍ \end{array} \right.␍ \; (j \ne i),␍ \qquad␍ \bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
||||
\bar{a}_{ij}^k =
|
||||
\left \{ \begin{array}{ll}
|
||||
a_{ij}^k & \mbox{if } j \in \mathcal{N}_i^k(\theta), \\
|
||||
0 & \mbox{otherwise},
|
||||
\end{array} \right.
|
||||
\; (j \ne i),
|
||||
\qquad
|
||||
\bar{a}_{ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
@@ -270,13 +329,15 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
<TD WIDTH=10 ALIGN="RIGHT">
|
||||
(5)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍and <IMG
|
||||
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
and <IMG
|
||||
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img44.png"
|
||||
ALT="$\omega^k$"> is an approximation of <IMG
|
||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img45.png"
|
||||
ALT="$4/(3\rho^k)$">, where␍<IMG
|
||||
ALT="$4/(3\rho^k)$">, where
|
||||
<IMG
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
|
||||
@@ -286,25 +347,32 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img47.png"
|
||||
ALT="$(D^k)^{-1}A^k_F$"> [<A
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>].␍In MLD2P4 this approximation is obtained by using <!-- MATH
|
||||
HREF="node29.html#BREZINA_VANEK">2</A>].
|
||||
In MLD2P4 this approximation is obtained by using <!-- MATH
|
||||
$\| A^k_F \|_\infty$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate␍of <IMG
|
||||
ALT="$\Vert A^k_F \Vert _\infty$"> as an estimate
|
||||
of <IMG
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic␍problems, filtering the matrix <IMG
|
||||
ALT="$\rho^k$">. Note that for systems coming from uniformly elliptic
|
||||
problems, filtering the matrix <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> has little or no effect, and␍<IMG
|
||||
ALT="$A^k$"> has little or no effect, and
|
||||
<IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> can be used instead of <IMG
|
||||
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img49.png"
|
||||
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.␍␍</FONT></FONT></FONT><HR>
|
||||
ALT="$A^k_F$">. The latter choice is the default in MLD2P4.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html241"
|
||||
HREF="node14.html">
|
||||
|
||||
+80
-27
@@ -53,30 +53,49 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
|
||||
<BR>
|
||||
Smoothers and coarsest-level solvers
|
||||
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,␍a hybrid version of the forward and backward Gauss-Seidel methods, and the␍additive Schwarz (AS) ones (see, e.g., [<A
|
||||
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
|
||||
a hybrid version of the forward and backward Gauss-Seidel methods, and the
|
||||
additive Schwarz (AS) ones (see, e.g., [<A
|
||||
HREF="node29.html#Saad_book">20</A>,<A
|
||||
HREF="node29.html#dd2_96">21</A>]). ␍␍The hybrid Gauss-Seidel␍version is considered because the original Gauss-Seidel method is inherently sequential.␍At each iteration of the hybrid version, each parallel process uses the most recent values␍of its own local variables and the values of the non-local variables computed at the␍previous iteration, obtained by exchanging data with other processes before␍the beginning of the current iteration.␍␍In the AS methods, the index space <IMG
|
||||
HREF="node29.html#dd2_96">21</A>]).
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The hybrid Gauss-Seidel
|
||||
version is considered because the original Gauss-Seidel method is inherently sequential.
|
||||
At each iteration of the hybrid version, each parallel process uses the most recent values
|
||||
of its own local variables and the values of the non-local variables computed at the
|
||||
previous iteration, obtained by exchanging data with other processes before
|
||||
the beginning of the current iteration.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In the AS methods, the index space <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$"> is divided into <IMG
|
||||
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img50.png"
|
||||
ALT="$m_k$">␍subsets <IMG
|
||||
ALT="$m_k$">
|
||||
subsets <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$\Omega^k_i$"> of size <IMG
|
||||
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img52.png"
|
||||
ALT="$n_{k,i}$">, possibly␍overlapping. For each <IMG
|
||||
ALT="$n_{k,i}$">, possibly
|
||||
overlapping. For each <IMG
|
||||
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i$"> we consider the restriction␍operator <!-- MATH
|
||||
ALT="$i$"> we consider the restriction
|
||||
operator <!-- MATH
|
||||
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img53.png"
|
||||
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">␍that maps a vector <IMG
|
||||
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
|
||||
that maps a vector <IMG
|
||||
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$x^k$"> to the vector <IMG
|
||||
@@ -85,16 +104,19 @@ Smoothers and coarsest-level solvers
|
||||
ALT="$x_i^k$"> made of the components of <IMG
|
||||
WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img54.png"
|
||||
ALT="$x^k$">␍with indices in <IMG
|
||||
ALT="$x^k$">
|
||||
with indices in <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$\Omega^k_i$">, and the prolongation operator␍<!-- MATH
|
||||
ALT="$\Omega^k_i$">, and the prolongation operator
|
||||
<!-- MATH
|
||||
$P^k_i = (R_i^k)^T$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img56.png"
|
||||
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build␍<!-- MATH
|
||||
ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
|
||||
<!-- MATH
|
||||
$A_i^k=R_i^kA^kP_i^k$
|
||||
-->
|
||||
<IMG
|
||||
@@ -103,13 +125,16 @@ Smoothers and coarsest-level solvers
|
||||
ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> to the index␍space <IMG
|
||||
ALT="$A^k$"> to the index
|
||||
space <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img51.png"
|
||||
ALT="$\Omega^k_i$">.␍The classical AS preconditioner <IMG
|
||||
ALT="$\Omega^k_i$">.
|
||||
The classical AS preconditioner <IMG
|
||||
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="$M^k_{AS}$"> is defined as␍</FONT></FONT></FONT>
|
||||
ALT="$M^k_{AS}$"> is defined as
|
||||
</FONT></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
@@ -124,25 +149,30 @@ Smoothers and coarsest-level solvers
|
||||
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <IMG
|
||||
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
where <IMG
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate␍inverse of <IMG
|
||||
ALT="$A_i^k$"> is supposed to be nonsingular. We observe that an approximate
|
||||
inverse of <IMG
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$A_i^k$"> is usually considered instead of <IMG
|
||||
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img61.png"
|
||||
ALT="$(A_i^k)^{-1}$">.␍The setup of <IMG
|
||||
ALT="$(A_i^k)^{-1}$">.
|
||||
The setup of <IMG
|
||||
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="$M^k_{AS}$"> during the multilevel build phase␍involves␍</FONT></FONT></FONT>
|
||||
|
||||
ALT="$M^k_{AS}$"> during the multilevel build phase
|
||||
involves
|
||||
</FONT></FONT></FONT>
|
||||
<UL>
|
||||
<LI>the definition of the index subspaces <IMG
|
||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img62.png"
|
||||
ALT="$\Omega_i^k$"> and of the corresponding ␍ operators <IMG
|
||||
ALT="$\Omega_i^k$"> and of the corresponding
|
||||
operators <IMG
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img63.png"
|
||||
ALT="$R_i^k$"> (and <IMG
|
||||
@@ -155,9 +185,11 @@ Smoothers and coarsest-level solvers
|
||||
SRC="img60.png"
|
||||
ALT="$A_i^k$">;
|
||||
</LI>
|
||||
<LI>the computation of their inverses (usually approximated␍ through some form of incomplete factorization).
|
||||
<LI>the computation of their inverses (usually approximated
|
||||
through some form of incomplete factorization).
|
||||
</LI>
|
||||
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍The computation of <!-- MATH
|
||||
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
The computation of <!-- MATH
|
||||
$z^k=M^k_{AS}w^k$
|
||||
-->
|
||||
<IMG
|
||||
@@ -169,8 +201,9 @@ Smoothers and coarsest-level solvers
|
||||
<IMG
|
||||
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img66.png"
|
||||
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the␍multilevel application phase, requires␍</FONT></FONT></FONT>
|
||||
|
||||
ALT="$w^k \in \mathbb{R}^{n_k}$">, during the
|
||||
multilevel application phase, requires
|
||||
</FONT></FONT></FONT>
|
||||
<UL>
|
||||
<LI>the restriction of <IMG
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
@@ -181,7 +214,8 @@ Smoothers and coarsest-level solvers
|
||||
<IMG
|
||||
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="$\mathbb{R}^{n_{k,i}}$">,␍ i.e. <!-- MATH
|
||||
ALT="$\mathbb{R}^{n_{k,i}}$">,
|
||||
i.e. <!-- MATH
|
||||
$w_i^k = R_i^{k} w^k$
|
||||
-->
|
||||
<IMG
|
||||
@@ -197,7 +231,8 @@ Smoothers and coarsest-level solvers
|
||||
SRC="img70.png"
|
||||
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$">;
|
||||
</LI>
|
||||
<LI>the prolongation and the sum of the previous vectors,␍ i.e. <!-- MATH
|
||||
<LI>the prolongation and the sum of the previous vectors,
|
||||
i.e. <!-- MATH
|
||||
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
|
||||
-->
|
||||
<IMG
|
||||
@@ -205,9 +240,27 @@ Smoothers and coarsest-level solvers
|
||||
SRC="img71.png"
|
||||
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$">.
|
||||
</LI>
|
||||
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍Variants of the classical AS method, which use modifications of the␍restriction and prolongation operators, are also implemented in MLD2P4.␍Among them, the Restricted AS (RAS) preconditioner usually␍outperforms the classical AS preconditioner in terms of convergence␍rate and of computation and communication time on parallel distributed-memory␍computers, and is therefore the most widely used among the AS␍preconditioners [<A
|
||||
HREF="node29.html#CAI_SARKIS">6</A>]. ␍␍Direct solvers based on sparse LU factorizations, implemented in the␍third-party libraries reported in Section <A HREF="node7.html#sec:third-party">3.2</A>, can be applied␍as coarsest-level solvers by MLD2P4. Native inexact solvers based on␍incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,␍and block Jacobi preconditioners are also available. Direct solvers usually␍lead to more effective preconditioners in terms of algorithmic scalability;␍however, this does not guarantee parallel efficiency.␍
|
||||
</FONT></FONT></FONT><HR>
|
||||
</UL><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
|
||||
Variants of the classical AS method, which use modifications of the
|
||||
restriction and prolongation operators, are also implemented in MLD2P4.
|
||||
Among them, the Restricted AS (RAS) preconditioner usually
|
||||
outperforms the classical AS preconditioner in terms of convergence
|
||||
rate and of computation and communication time on parallel distributed-memory
|
||||
computers, and is therefore the most widely used among the AS
|
||||
preconditioners [<A
|
||||
HREF="node29.html#CAI_SARKIS">6</A>].
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Direct solvers based on sparse LU factorizations, implemented in the
|
||||
third-party libraries reported in Section <A HREF="node7.html#sec:third-party">3.2</A>, can be applied
|
||||
as coarsest-level solvers by MLD2P4. Native inexact solvers based on
|
||||
incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
|
||||
and block Jacobi preconditioners are also available. Direct solvers usually
|
||||
lead to more effective preconditioners in terms of algorithmic scalability;
|
||||
however, this does not guarantee parallel efficiency.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT></FONT><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html251"
|
||||
HREF="node15.html">
|
||||
|
||||
@@ -122,7 +122,7 @@ Examples showing the basic use of MLD2P4 are reported in Section <A HREF="n
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="857"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="897"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 1:</STRONG>
|
||||
Preconditioner types, corresponding strings and default choices.
|
||||
|
||||
@@ -88,7 +88,7 @@ the corresponding codes are available in <code>examples/fileread/</code>.
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex1"></A><A NAME="860"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex1"></A><A NAME="900"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2:</STRONG>
|
||||
setup and application of the default multi-level preconditioner (example 1).
|
||||
@@ -194,7 +194,7 @@ boundary conditions are also available in the directory <code>examples/pdegen</c
|
||||
</FONT></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex2"></A><A NAME="862"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex2"></A><A NAME="902"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3:</STRONG>
|
||||
setup of a multi-level preconditioner</CAPTION>
|
||||
@@ -227,7 +227,7 @@ setup of a multi-level preconditioner</CAPTION>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex3"></A><A NAME="864"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex3"></A><A NAME="904"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4:</STRONG>
|
||||
setup of a multi-level preconditioner</CAPTION>
|
||||
@@ -260,7 +260,7 @@ setup of a multi-level preconditioner</CAPTION>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex4"></A><A NAME="866"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:ex4"></A><A NAME="906"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5:</STRONG>
|
||||
setup of a one-level Schwarz preconditioner.</CAPTION>
|
||||
|
||||
@@ -249,7 +249,7 @@ solver is changed to the default sequential solver.
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1290"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1330"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 2:</STRONG>
|
||||
Parameters defining the multi-level cycle and the number of cycles to
|
||||
@@ -302,7 +302,7 @@ number <IMG
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1295"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1335"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 3:</STRONG>
|
||||
Parameters defining the aggregation algorithm.
|
||||
@@ -417,7 +417,7 @@ of levels.</TD>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1299"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1339"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 4:</STRONG>
|
||||
Parameters defining the aggregation algorithm (continued).
|
||||
@@ -484,7 +484,7 @@ the parameter <TT>ilev</TT>.</TD>
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1304"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1344"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 5:</STRONG>
|
||||
Parameters defining the coarse-space correction at the coarsest
|
||||
@@ -592,7 +592,7 @@ Note that <TT>UMF</TT> and <TT>SLU</TT> require the coarsest
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
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<BR><P></P>
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<DIV ALIGN="CENTER"><A NAME="1306"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1346"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 6:</STRONG>
|
||||
Parameters defining the coarse-space correction at the coarsest
|
||||
@@ -658,7 +658,7 @@ number <IMG
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1308"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1348"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 7:</STRONG>
|
||||
Parameters defining the smoother or the details of the one-level preconditioner.
|
||||
@@ -781,7 +781,7 @@ Parameters defining the smoother or the details of the one-level preconditioner.
|
||||
<P>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER"><A NAME="1310"></A>
|
||||
<DIV ALIGN="CENTER"><A NAME="1350"></A>
|
||||
<TABLE>
|
||||
<CAPTION><STRONG>Table 8:</STRONG>
|
||||
Parameters defining the smoother or the details of the one-level preconditioner
|
||||
|
||||
@@ -9981,8 +9981,8 @@ endobj
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680 0 obj
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<<
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||||
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/CreationDate (D:20170725140354+01'00')
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/PTEX.Fullbanner (This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016) kpathsea version 6.2.2)
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>>
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/Root 679 0 R
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/Length 3410
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>>
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