mirror of
https://github.com/sfilippone/amg4psblas.git
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Fixup the documentation.
This commit is contained in:
Salvatore Filippone
+60
-47
@@ -1,6 +1,6 @@
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
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<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
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<HTML>
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<HEAD>
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<TITLE>Smoothed Aggregation</TITLE>
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@@ -9,7 +9,7 @@
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<META NAME="resource-type" CONTENT="document">
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<META NAME="distribution" CONTENT="global">
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<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
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<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
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<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
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<LINK REL="STYLESHEET" HREF="userhtml.css">
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@@ -54,27 +54,27 @@ Smoothed Aggregation
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</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN>, used to compute
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the coarse-level matrix <SPAN CLASS="MATH"><IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
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algorithm described in [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<A
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HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>].
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HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>].
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The basic idea of this algorithm is to build a coarse set of indices
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<SPAN CLASS="MATH"><IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"></SPAN> into disjoint
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subsets (aggregates), and to define the coarse-to-fine space transfer operator
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<SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN> 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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@@ -84,26 +84,26 @@ prolongation operator, with the aim of improving the quality of the coarse-space
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</BIG></BIG></BIG>
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<OL>
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<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img26.png"
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ALT="$\Omega^{k+1}$"></SPAN>;
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</LI>
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<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN>;
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</LI>
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<LI>application of <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><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="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
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WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img15.png"
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ALT="$A^{k+1}$"></SPAN>.
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</LI>
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@@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
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described in [<A
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HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
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HREF="node36.html#VANEK_MANDEL_BREZINA">26</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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<SPAN CLASS="MATH"><IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"></SPAN>,
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consisting of a suitably chosen index <!-- MATH
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@@ -133,7 +133,7 @@ consisting of a suitably chosen index <!-- MATH
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SRC="img29.png"
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ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
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strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
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WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img30.png"
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ALT="$i$"></SPAN>, i.e.,
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</BIG></BIG></BIG>
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@@ -149,11 +149,13 @@ strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
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<A NAME="eq:strongly_coup"></A>
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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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WIDTH="387" HEIGHT="49" 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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\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
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\left\{ r \i...
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...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
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</TABLE>
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@@ -162,10 +164,10 @@ for a given threshold <!-- MATH
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$\theta \in [0,1]$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img32.png"
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ALT="$\theta \in [0,1]$"></SPAN> (see [<A
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HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
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HREF="node36.html#VANEK_MANDEL_BREZINA">26</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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@@ -180,11 +182,11 @@ MLD2P4, since it has been shown to produce good results in practice
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[<A
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HREF="node36.html#aaecc_07">5</A>,<A
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HREF="node36.html#apnum_07">7</A>,<A
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HREF="node36.html#TUMINARO_TONG">24</A>].
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HREF="node36.html#TUMINARO_TONG">25</A>].
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</BIG></BIG></BIG>
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<P>
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<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
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<!-- MATH
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@@ -210,10 +212,14 @@ MLD2P4, since it has been shown to produce good results in practice
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<A NAME="eq:tent_prol"></A>
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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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WIDTH="286" HEIGHT="52" 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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ALT="\begin{displaymath}
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\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
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\left\{...
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...Omega^k_j, \\
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0 & \quad \mbox{otherwise},
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\end{array} \right.
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
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@@ -223,21 +229,21 @@ where <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.png"
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ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img9.png"
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ALT="$\Omega^k$"></SPAN>
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corresponding to the index <!-- MATH
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$j \in \Omega^{k+1}$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img27.png"
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ALT="$j \in \Omega^{k+1}$"></SPAN>.
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<SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img35.png"
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ALT="$\bar{P}^k$"></SPAN> a smoother
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<!-- MATH
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@@ -257,9 +263,11 @@ P^k = S^k \bar{P}^k,
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-->
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<IMG
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WIDTH="90" HEIGHT="30" BORDER="0"
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WIDTH="91" HEIGHT="30" BORDER="0"
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SRC="img37.png"
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ALT="\begin{displaymath}␍P^k = S^k \bar{P}^k,␍\end{displaymath}">
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ALT="\begin{displaymath}
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P^k = S^k \bar{P}^k,
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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@@ -267,9 +275,9 @@ in order to remove nonsmooth components from the range of the prolongator,
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and hence to improve the convergence properties of the multilevel
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method [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<A
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HREF="node36.html#Stuben_01">23</A>].
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HREF="node36.html#Stuben_01">24</A>].
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A simple choice for <SPAN CLASS="MATH"><IMG
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
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SRC="img38.png"
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ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
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</BIG></BIG></BIG>
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@@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
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-->
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<IMG
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WIDTH="175" HEIGHT="31" BORDER="0"
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WIDTH="176" HEIGHT="32" BORDER="0"
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SRC="img39.png"
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ALT="\begin{displaymath}␍S^k = I - \omega^k (D^k)^{-1} A^k_F , ␍\end{displaymath}">
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ALT="\begin{displaymath}
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S^k = I - \omega^k (D^k)^{-1} A^k_F ,
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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where <SPAN CLASS="MATH"><IMG
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WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$A^k$"></SPAN>,
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<!-- MATH
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$A^k_F = (\bar{a}_{ij}^k)$
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
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</BIG></BIG></BIG>
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@@ -321,17 +331,20 @@ where <SPAN CLASS="MATH"><IMG
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<A NAME="eq:filtered"></A>
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
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WIDTH="514" HEIGHT="74" BORDER="0"
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WIDTH="499" HEIGHT="59" BORDER="0"
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SRC="img43.png"
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ALT="\begin{displaymath}
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\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & ...
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...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
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\bar{a}_{ij}^k =
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\left \{ \begin{array}{ll}
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a_{ij}^k & \m...
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...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
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\end{displaymath}"></TD>
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<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
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(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
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and <SPAN CLASS="MATH"><IMG
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WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
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SRC="img44.png"
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ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
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WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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@@ -360,14 +373,14 @@ of <SPAN CLASS="MATH"><IMG
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SRC="img46.png"
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ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
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problems, filtering the matrix <SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$A^k$"></SPAN> has little or no effect, and
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<SPAN CLASS="MATH"><IMG
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WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
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WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img49.png"
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ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
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</BIG></BIG></BIG>
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