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Fixed doc copyright vs contribution
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Salvatore Filippone
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@@ -7,8 +7,8 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<HTML>
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<HEAD>
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<TITLE>AMG preconditioners</TITLE>
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<META NAME="description" CONTENT="AMG preconditioners">
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<TITLE>Multigrid Background</TITLE>
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@@ -18,293 +18,146 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> Next:</B> <A NAME="tex2html230"
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HREF="node13.html">Smoothed Aggregation</A>
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<B> Previous:</B> <A NAME="tex2html217"
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HREF="node11.html">Example and test programs</A>
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<B> <A NAME="tex2html225"
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HREF="node2.html">Contents</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
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<H1><A NAME="SECTION00060000000000000000"></A><A NAME="sec:background"></A>
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<BR>
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AMG preconditioners
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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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Multigrid Background
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</H1><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
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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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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
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WIDTH="58" 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 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">
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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;
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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
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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.,
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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="$\Omega = \{1, 2, \ldots, n\}$">.
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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 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,
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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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<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},
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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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<IMG
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WIDTH="398" HEIGHT="30" BORDER="0"
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SRC="img7.png"
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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">
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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
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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.
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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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WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img8.png"
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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="$\Omega^k$">,
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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="$\Omega^k$">.
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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,
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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="$k+1$">:
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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
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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="node30.html#Briggs2000">3</A>,<A
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HREF="node30.html#Stuben_01">23</A>,<A
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HREF="node30.html#dd2_96">21</A>] for details.)
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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
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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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<IMG
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WIDTH="254" HEIGHT="30" BORDER="0"
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SRC="img14.png"
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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">
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the matrix <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$"> is computed by using the previous operators according
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to the Galerkin approach, i.e.,
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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="node30.html#BREZINA_VANEK">2</A>,<A
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HREF="node30.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="node30.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="node13.html#sec:multilevel">4.1</A>-<A HREF="node15.html#sec:smoothers">4.3</A>. For further details the reader
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is referred to [<A
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HREF="node30.html#para_04">4</A>,<A
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HREF="node30.html#aaecc_07">5</A>,<A
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HREF="node30.html#apnum_07">7</A>,<A
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HREF="node30.html#MLD2P4_TOMS">8</A>].
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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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A^{k+1}=R^kA^kP^k.
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="27" BORDER="0"
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SRC="img16.png"
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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">
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In the current implementation of MLD2P4 we have <IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="$R^k=(P^k)^T$">
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A smoother with iteration matrix <IMG
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WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img18.png"
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ALT="$M^k$"> is set up at each level <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="$k < nlev$">, and a solver
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is set up at the coarsest level, so that they are ready for application
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(for example, setting up a solver based on the <IMG
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WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img19.png"
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ALT="$LU$"> factorization means computing
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and storing the <IMG
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WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$L$"> and <IMG
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WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$U$"> factors). The construction of the hierarchy of AMG components
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described so far corresponds to the so-called build phase of the preconditioner.
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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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<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
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Application phase of a V-cycle preconditioner.</CAPTION>
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<TR><TD>
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<DIV ALIGN="CENTER">
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<!-- MATH
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$\framebox{
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\begin{minipage}{.85\textwidth}
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\begin{tabbing}
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\quad \=\quad \=\quad \=\quad \\[-3mm]
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procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
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\>if $\left(k \ne nlev \right)$\ then \\[1mm]
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\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
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\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
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\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
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\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
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\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
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\>else \\[1mm]
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\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
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\>endif \\[1mm]
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\>return $u^k$\ \\[1mm]
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||||
end
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\end{tabbing}
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\end{minipage}
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}$
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-->
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<IMG
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WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
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SRC="img22.png"
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ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
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...mm]␍\>endif \ [1mm]␍\>return $u^k$ \ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
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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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||||
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||||
</DIV></TD></TR>
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</TABLE>
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</DIV>
|
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<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">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
|
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and W-cycle [<A
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HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
|
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in [<A
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||||
HREF="node29.html#Notay2008">19</A>].
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</FONT></FONT></FONT>
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||||
<P>
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||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
|
||||
<UL>
|
||||
<LI><A NAME="tex2html228"
|
||||
HREF="node13.html">AMG preconditioners</A>
|
||||
<LI><A NAME="tex2html229"
|
||||
HREF="node14.html">Smoothed Aggregation</A>
|
||||
<LI><A NAME="tex2html230"
|
||||
HREF="node15.html">Smoothers and coarsest-level solvers</A>
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HREF="node11.html">Multigrid Background</A>
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Reference in New Issue
Block a user