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Fixed docs.

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Salvatore Filippone
2018-05-14 14:50:52 +01:00
parent 4b43164668
commit b7e8a921d8
101 changed files with 85 additions and 85 deletions
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docs/html/node14.html
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@@ -54,11 +54,11 @@ Smoothed Aggregation
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN>, used to compute
the coarse-level matrix <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
algorithm described in [<A
@@ -66,15 +66,15 @@ algorithm described in [<A
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>].
The basic idea of this algorithm is to build a coarse set of indices
<SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN> into disjoint
subsets (aggregates), and to define the coarse-to-fine space transfer operator
<SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
prolongation operator, with the aim of improving the quality of the coarse-space correction.
@@ -84,26 +84,26 @@ prolongation operator, with the aim of improving the quality of the coarse-space
</BIG></BIG></BIG>
<OL>
<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img26.png"
ALT="$\Omega^{k+1}$"></SPAN>;
</LI>
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN>;
</LI>
<LI>application of <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$"></SPAN>.
</LI>
@@ -116,13 +116,13 @@ each index <!-- MATH
$j \in \Omega^{k+1}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN>,
consisting of a suitably chosen index <!-- MATH
@@ -133,7 +133,7 @@ consisting of a suitably chosen index <!-- MATH
SRC="img29.png"
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img30.png"
ALT="$i$"></SPAN>, i.e.,
</BIG></BIG></BIG>
@@ -149,7 +149,7 @@ strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
<A NAME="eq:strongly_coup"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
WIDTH="387" HEIGHT="49" BORDER="0"
WIDTH="387" HEIGHT="48" BORDER="0"
SRC="img31.png"
ALT="\begin{displaymath}
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
@@ -164,7 +164,7 @@ for a given threshold <!-- MATH
$\theta \in [0,1]$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img32.png"
ALT="$\theta \in [0,1]$"></SPAN> (see&nbsp;[<A
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] for the details).
@@ -175,7 +175,7 @@ distribution. This version is embarrassingly parallel, since it does not require
communication. On the other hand, it may produce some nonuniform aggregates
and is strongly dependent on the number of processors and on the initial partitioning
of the matrix <SPAN CLASS="MATH"><IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"></SPAN>. Nevertheless, this parallel algorithm has been chosen for
MLD2P4, since it has been shown to produce good results in practice
@@ -186,7 +186,7 @@ MLD2P4, since it has been shown to produce good results in practice
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
<!-- MATH
@@ -212,7 +212,7 @@ MLD2P4, since it has been shown to produce good results in practice
<A NAME="eq:tent_prol"></A>
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
WIDTH="286" HEIGHT="52" BORDER="0"
WIDTH="286" HEIGHT="51" BORDER="0"
SRC="img34.png"
ALT="\begin{displaymath}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
@@ -229,21 +229,21 @@ where <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img28.png"
ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN>
corresponding to the index <!-- MATH
$j \in \Omega^{k+1}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img27.png"
ALT="$j \in \Omega^{k+1}$"></SPAN>.
<SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img35.png"
ALT="$\bar{P}^k$"></SPAN> a smoother
<!-- MATH
@@ -263,7 +263,7 @@ P^k = S^k \bar{P}^k,
-->
<IMG
WIDTH="91" HEIGHT="30" BORDER="0"
WIDTH="90" HEIGHT="30" BORDER="0"
SRC="img37.png"
ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
@@ -277,7 +277,7 @@ method&nbsp;[<A
HREF="node36.html#BREZINA_VANEK">2</A>,<A
HREF="node36.html#Stuben_01">24</A>].
A simple choice for <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img38.png"
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
</BIG></BIG></BIG>
@@ -290,7 +290,7 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
-->
<IMG
WIDTH="176" HEIGHT="32" BORDER="0"
WIDTH="175" HEIGHT="31" BORDER="0"
SRC="img39.png"
ALT="\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
@@ -299,17 +299,17 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
<BR CLEAR="ALL">
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
where <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"></SPAN>,
<!-- MATH
$A^k_F = (\bar{a}_{ij}^k)$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
</BIG></BIG></BIG>
@@ -344,7 +344,7 @@ a_{ij}^k &amp; \m...
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
and <SPAN CLASS="MATH"><IMG
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img44.png"
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
@@ -373,14 +373,14 @@ of <SPAN CLASS="MATH"><IMG
SRC="img46.png"
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
problems, filtering the matrix <SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"></SPAN> has little or no effect, and
<SPAN CLASS="MATH"><IMG
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img49.png"
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
</BIG></BIG></BIG>