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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>Smoothers and coarsest-level solvers</TITLE>
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<META NAME="description" CONTENT="Smoothers and coarsest-level solvers">
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<TITLE>Smoothed Aggregation</TITLE>
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<META NAME="description" CONTENT="Smoothed Aggregation">
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<META NAME="resource-type" CONTENT="document">
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<META NAME="distribution" CONTENT="global">
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@@ -18,270 +18,382 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<BR>
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<B> Next:</B> <A NAME="tex2html252"
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HREF="node15.html">Getting Started</A>
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HREF="node11.html">Multigrid Background</A>
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HREF="node15.html">Smoothers and coarsest-level solvers</A>
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HREF="node12.html">Multigrid Background</A>
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HREF="node13.html">Smoothed Aggregation</A>
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<B> <A NAME="tex2html250"
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HREF="node13.html">AMG preconditioners</A>
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<B> <A NAME="tex2html252"
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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="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
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<H2><A NAME="SECTION00062000000000000000"></A><A NAME="sec:aggregation"></A>
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<BR>
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Smoothers and coarsest-level solvers
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Smoothed Aggregation
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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">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
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a hybrid version of the forward and backward Gauss-Seidel methods, and the
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additive Schwarz (AS) ones (see, e.g., [<A
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HREF="node29.html#Saad_book">20</A>,<A
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HREF="node29.html#dd2_96">21</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">The hybrid Gauss-Seidel
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version is considered because the original Gauss-Seidel method is inherently sequential.
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At each iteration of the hybrid version, each parallel process uses the most recent values
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of its own local variables and the values of the non-local variables computed at the
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previous iteration, obtained by exchanging data with other processes before
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the beginning of the current iteration.
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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">In the AS methods, the index space <IMG
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<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to define the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">, used to compute
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the coarse-level 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="$A^{k+1}$">, MLD2P4 uses the smoothed aggregation
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algorithm described in [<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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The basic idea of this algorithm is to build a coarse set of indices
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<IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"> by suitably grouping the indices 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="$\Omega^k$"> is divided into <IMG
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WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img50.png"
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ALT="$m_k$">
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subsets <IMG
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ALT="$\Omega^k$"> into disjoint
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subsets (aggregates), and to define the coarse-to-fine space transfer operator
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<IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> by applying a suitable smoother to a simple piecewise constant
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prolongation operator, with the aim of improving the quality of the coarse-space correction.
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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">Three main steps can be identified in the smoothed aggregation procedure:
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</FONT></FONT></FONT>
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<OL>
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<LI>aggregation of the indices 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="$\Omega^k$"> to obtain <IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$">;
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</LI>
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<LI>construction of the prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$">;
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</LI>
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<LI>application of <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"> and <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="$R^k=(P^k)^T$"> to build <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="$A^{k+1}$">.
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</LI>
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</OL><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 perform the coarsening step, the smoothed aggregation algorithm
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described in [<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
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each index <!-- MATH
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$j \in \Omega^{k+1}$
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-->
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<IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"> corresponds to an aggregate <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$\Omega^k_i$"> of size <IMG
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WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img52.png"
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ALT="$n_{k,i}$">, possibly
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overlapping. For each <IMG
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SRC="img28.png"
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ALT="$\Omega^k_j$"> 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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consisting of a suitably chosen index <!-- MATH
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$i \in \Omega^k$
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-->
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<IMG
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WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$i \in \Omega^k$"> and indices that are (usually) contained in a
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strongly-coupled neighborood of <IMG
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WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img30.png"
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ALT="$i$"> we consider the restriction
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operator <!-- MATH
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$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
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-->
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<IMG
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WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img53.png"
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ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$">
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that maps a vector <IMG
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WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img54.png"
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ALT="$x^k$"> to the vector <IMG
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WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img55.png"
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ALT="$x_i^k$"> made of the components of <IMG
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WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img54.png"
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ALT="$x^k$">
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with indices in <IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img51.png"
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ALT="$\Omega^k_i$">, and the prolongation operator
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ALT="$i$">, i.e.,
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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$P^k_i = (R_i^k)^T$
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\begin{equation}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
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\left\{ r \in \Omega^k: |a_{ir}^k| > \theta \sqrt{|a_{ii}^ka_{rr}^k|} \right \} \cup \left\{ i \right\},
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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:strongly_coup"></A><IMG
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WIDTH="387" HEIGHT="72" BORDER="0"
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SRC="img31.png"
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ALT="\begin{displaymath}
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\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
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...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(3)</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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for a given threshold <!-- MATH
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$\theta \in [0,1]$
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-->
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<IMG
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WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img56.png"
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ALT="$P^k_i = (R_i^k)^T$">. These operators are then used to build
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<!-- MATH
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$A_i^k=R_i^kA^kP_i^k$
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-->
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<IMG
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WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
|
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ALT="$A_i^k=R_i^kA^kP_i^k$">, which is the restriction of <IMG
|
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WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img32.png"
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ALT="$\theta \in [0,1]$"> (see [<A
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HREF="node30.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
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Since this algorithm has a sequential nature, a decoupled
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version of it is applied, where each processor independently executes
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the algorithm on the set of indices assigned to it in the initial data
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distribution. This version is embarrassingly parallel, since it does not require any data
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communication. On the other hand, it may produce some nonuniform aggregates
|
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and is strongly dependent on the number of processors and on the initial partitioning
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of the matrix <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="$A$">. Nevertheless, this parallel algorithm has been chosen for
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MLD2P4, since it has been shown to produce good results in practice
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[<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#TUMINARO_TONG">24</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">The prolongator <IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$A^k$"> to the index
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space <IMG
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SRC="img25.png"
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ALT="$P^k$"> is built starting from a tentative prolongator
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<!-- MATH
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$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$
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-->
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<IMG
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WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img33.png"
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ALT="$\bar{P}^k \in \mathbb{R}^{n_k \times n_{k+1}}$">, defined as
|
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</FONT></FONT></FONT>
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
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\left\{ \begin{array}{ll}
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1 & \quad \mbox{if} \; i \in \Omega^k_j, \\
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0 & \quad \mbox{otherwise},
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\end{array} \right.
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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:tent_prol"></A><IMG
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WIDTH="287" HEIGHT="51" BORDER="0"
|
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SRC="img34.png"
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ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
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...ega^k_j, ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(4)</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 <IMG
|
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img51.png"
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ALT="$\Omega^k_i$">.
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The classical AS preconditioner <IMG
|
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WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img58.png"
|
||||
ALT="$M^k_{AS}$"> is defined as
|
||||
SRC="img28.png"
|
||||
ALT="$\Omega^k_j$"> is the aggregate of <IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$">
|
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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
|
||||
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
|
||||
$S^k \in \mathbb{R}^{n_k \times n_k}$
|
||||
-->
|
||||
<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>
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||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
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||||
\begin{displaymath}
|
||||
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
|
||||
P^k = S^k \bar{P}^k,
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="219" HEIGHT="59" BORDER="0"
|
||||
SRC="img59.png"
|
||||
ALT="\begin{displaymath}␍ ( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},␍\end{displaymath}">
|
||||
WIDTH="90" HEIGHT="30" BORDER="0"
|
||||
SRC="img37.png"
|
||||
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
|
||||
HREF="node30.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node30.html#Stuben_01">23</A>].
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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>
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<BR><P></P>
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<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
\end{displaymath}
|
||||
-->
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||||
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<IMG
|
||||
WIDTH="175" HEIGHT="31" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
|
||||
</DIV>
|
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<BR CLEAR="ALL">
|
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<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
|
||||
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
|
||||
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img58.png"
|
||||
ALT="$M^k_{AS}$"> during the multilevel build phase
|
||||
involves
|
||||
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
|
||||
$A^k_F = (\bar{a}_{ij}^k)$
|
||||
-->
|
||||
<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>
|
||||
<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
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img63.png"
|
||||
ALT="$R_i^k$"> (and <IMG
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img64.png"
|
||||
ALT="$P_i^k$">);
|
||||
</LI>
|
||||
<LI>the computation of the submatrices <IMG
|
||||
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img60.png"
|
||||
ALT="$A_i^k$">;
|
||||
</LI>
|
||||
<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
|
||||
$z^k=M^k_{AS}w^k$
|
||||
<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),
|
||||
\end{equation}
|
||||
-->
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
||||
WIDTH="514" HEIGHT="74" BORDER="0"
|
||||
SRC="img43.png"
|
||||
ALT="\begin{displaymath}
|
||||
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & ...
|
||||
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
|
||||
<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
|
||||
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
|
||||
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img46.png"
|
||||
ALT="$\rho^k$"> is the spectral radius of <!-- MATH
|
||||
$(D^k)^{-1}A^k_F$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img65.png"
|
||||
ALT="$z^k=M^k_{AS}w^k$">, with <!-- MATH
|
||||
$w^k \in \mathbb{R}^{n_k}$
|
||||
WIDTH="83" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img47.png"
|
||||
ALT="$(D^k)^{-1}A^k_F$"> [<A
|
||||
HREF="node30.html#BREZINA_VANEK">2</A>].
|
||||
In MLD2P4 this approximation is obtained by using <!-- MATH
|
||||
$\| A^k_F \|_\infty$
|
||||
-->
|
||||
<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>
|
||||
<UL>
|
||||
<LI>the restriction of <IMG
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img67.png"
|
||||
ALT="$w^k$"> to the subspaces <!-- MATH
|
||||
$\mathbb{R}^{n_{k,i}}$
|
||||
-->
|
||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img48.png"
|
||||
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
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"> has little or no effect, and
|
||||
<IMG
|
||||
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img68.png"
|
||||
ALT="$\mathbb{R}^{n_{k,i}}$">,
|
||||
i.e. <!-- MATH
|
||||
$w_i^k = R_i^{k} w^k$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img69.png"
|
||||
ALT="$w_i^k = R_i^{k} w^k$">;
|
||||
</LI>
|
||||
<LI>the computation of the vectors <!-- MATH
|
||||
$z_i^k=(A_i^k)^{-1} w_i^k$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
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
|
||||
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
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>].
|
||||
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>
|
||||
<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>
|
||||
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html251"
|
||||
<A NAME="tex2html253"
|
||||
HREF="node15.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
||||
<A NAME="tex2html247"
|
||||
HREF="node11.html">
|
||||
<A NAME="tex2html249"
|
||||
HREF="node12.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
||||
<A NAME="tex2html243"
|
||||
HREF="node13.html">
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
||||
<A NAME="tex2html249"
|
||||
<A NAME="tex2html251"
|
||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html252"
|
||||
HREF="node15.html">Getting Started</A>
|
||||
<B> Up:</B> <A NAME="tex2html248"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> Next:</B> <A NAME="tex2html254"
|
||||
HREF="node15.html">Smoothers and coarsest-level solvers</A>
|
||||
<B> Up:</B> <A NAME="tex2html250"
|
||||
HREF="node12.html">Multigrid Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html244"
|
||||
HREF="node13.html">Smoothed Aggregation</A>
|
||||
<B> <A NAME="tex2html250"
|
||||
HREF="node13.html">AMG preconditioners</A>
|
||||
<B> <A NAME="tex2html252"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
|
||||
Reference in New Issue
Block a user