Fixup the documentation.
Salvatore Filippone
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@@ -69,17 +69,15 @@ based on PSBLAS</BIG></SPAN>
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<BR>
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<BR>
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<BR>
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Software version: 2.1
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Software version: 2.2
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<BR>
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July 31, 2017
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July 31, 2018
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<HTML>
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<HEAD>
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<TITLE>Abstract</TITLE>
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@@ -9,7 +9,7 @@
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<META NAME="resource-type" CONTENT="document">
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<HEAD>
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<TITLE>Bug reporting</TITLE>
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@@ -9,7 +9,7 @@
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<META NAME="resource-type" CONTENT="document">
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<TITLE>Example and test programs</TITLE>
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@@ -9,7 +9,7 @@
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<META NAME="resource-type" CONTENT="document">
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<TITLE>Multigrid Background</TITLE>
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@@ -9,7 +9,7 @@
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<META NAME="resource-type" CONTENT="document">
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@@ -88,19 +88,19 @@ are considered. The second approach performs a fully automatic coarsening and en
|
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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
|
||||
HREF="node36.html#Briggs2000">3</A>,<A
|
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HREF="node36.html#Stuben_01">23</A>,<A
|
||||
HREF="node36.html#dd2_96">21</A>] for details.)
|
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HREF="node36.html#Stuben_01">24</A>,<A
|
||||
HREF="node36.html#dd2_96">22</A>] for details.)
|
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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">MLD2P4 uses a pure algebraic approach, based on the smoothed
|
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aggregation algorithm [<A
|
||||
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>],
|
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HREF="node36.html#VANEK_MANDEL_BREZINA">26</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
|
||||
HREF="node36.html#TUMINARO_TONG">24</A>].
|
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HREF="node36.html#TUMINARO_TONG">25</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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<HTML>
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<HEAD>
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<TITLE>AMG preconditioners</TITLE>
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@@ -9,7 +9,7 @@
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@@ -67,7 +67,7 @@ Ax=b,
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<A NAME="eq:system"></A>
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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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WIDTH="57" 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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@@ -80,7 +80,7 @@ where <!-- MATH
|
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$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
|
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-->
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<SPAN CLASS="MATH"><IMG
|
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WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
|
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ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"></SPAN> is a nonsingular sparse matrix;
|
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for ease of presentation we assume <SPAN CLASS="MATH"><IMG
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@@ -98,7 +98,7 @@ pattern.
|
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$\Omega = \{1, 2, \ldots, n\}$
|
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-->
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<SPAN CLASS="MATH"><IMG
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WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img6.png"
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ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>.
|
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Any algebraic multilevel preconditioners implemented in MLD2P4 generates
|
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@@ -116,7 +116,8 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices,
|
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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}">
|
||||
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}">
|
||||
</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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@@ -132,28 +133,28 @@ A vector space <!-- MATH
|
||||
$\mathbb{R}^{n_{k}}$
|
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-->
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<SPAN CLASS="MATH"><IMG
|
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WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img8.png"
|
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ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$"></SPAN>,
|
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where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img10.png"
|
||||
ALT="$n_k$"></SPAN> is the size of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$"></SPAN>.
|
||||
For all <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$k < nlev$"></SPAN>, a restriction operator and a prolongation one are built,
|
||||
which connect two levels <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img12.png"
|
||||
ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img13.png"
|
||||
ALT="$k+1$"></SPAN>:
|
||||
</BIG></BIG></BIG>
|
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@@ -167,14 +168,17 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="254" HEIGHT="30" BORDER="0"
|
||||
WIDTH="255" HEIGHT="30" BORDER="0"
|
||||
SRC="img14.png"
|
||||
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}">
|
||||
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}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||
the matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$"></SPAN> is computed by using the previous operators according
|
||||
to the Galerkin approach, i.e.,
|
||||
@@ -188,9 +192,11 @@ A^{k+1}=R^kA^kP^k.
|
||||
-->
|
||||
|
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<IMG
|
||||
WIDTH="131" HEIGHT="27" BORDER="0"
|
||||
WIDTH="131" HEIGHT="28" BORDER="0"
|
||||
SRC="img16.png"
|
||||
ALT="\begin{displaymath}
A^{k+1}=R^kA^kP^k.
\end{displaymath}">
|
||||
ALT="\begin{displaymath}
|
||||
A^{k+1}=R^kA^kP^k.
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||
@@ -199,10 +205,10 @@ In the current implementation of MLD2P4 we have <SPAN CLASS="MATH"><IMG
|
||||
SRC="img17.png"
|
||||
ALT="$R^k=(P^k)^T$"></SPAN>
|
||||
A smoother with iteration matrix <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img18.png"
|
||||
ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img11.png"
|
||||
ALT="$k < nlev$"></SPAN>, and a solver
|
||||
is set up at the coarsest level, so that they are ready for application
|
||||
@@ -211,7 +217,7 @@ is set up at the coarsest level, so that they are ready for application
|
||||
SRC="img19.png"
|
||||
ALT="$LU$"></SPAN> factorization means computing
|
||||
and storing the <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$L$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
@@ -251,8 +257,15 @@ end
|
||||
<IMG
|
||||
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img22.png"
|
||||
ALT="\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad...
|
||||
...mm]
\>endif [1mm]
\>return $u^k$ [1mm]
end
\end{tabbing}
\end{minipage}
}">
|
||||
ALT="\framebox{
|
||||
\begin{minipage}{.85\textwidth}
|
||||
\begin{tabbing}
|
||||
\quad \=\quad \=\quad...
|
||||
...[1mm]
|
||||
\>endif [1mm]
|
||||
\>return $u^k$ [1mm]
|
||||
end
|
||||
\end{tabbing}\end{minipage}}">
|
||||
|
||||
</DIV></TD></TR>
|
||||
</TABLE>
|
||||
@@ -263,14 +276,14 @@ end
|
||||
to obtain different multilevel preconditioners;
|
||||
this is done in the application phase, i.e., in the computation of a vector
|
||||
of type <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img23.png"
|
||||
ALT="$w=B^{-1}v$"></SPAN>, where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img24.png"
|
||||
ALT="$B$"></SPAN> denotes the preconditioner, usually within an iteration
|
||||
of a Krylov solver [<A
|
||||
HREF="node36.html#Saad_book">20</A>]. An example of such a combination, known as
|
||||
HREF="node36.html#Saad_book">21</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
|
||||
@@ -278,7 +291,7 @@ performed. Other cycles can be defined; in MLD2P4, we implemented the standard V
|
||||
and W-cycle [<A
|
||||
HREF="node36.html#Briggs2000">3</A>], and a version of the K-cycle described
|
||||
in [<A
|
||||
HREF="node36.html#Notay2008">19</A>].
|
||||
HREF="node36.html#Notay2008">20</A>].
|
||||
</BIG></BIG></BIG>
|
||||
<P>
|
||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||
|
||||
@@ -1,6 +1,6 @@
|
||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
||||
|
||||
<!--Converted with LaTeX2HTML 2017.2 (Released Jan 23, 2017) -->
|
||||
<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
|
||||
<HTML>
|
||||
<HEAD>
|
||||
<TITLE>Smoothed Aggregation</TITLE>
|
||||
@@ -9,7 +9,7 @@
|
||||
<META NAME="resource-type" CONTENT="document">
|
||||
<META NAME="distribution" CONTENT="global">
|
||||
|
||||
<META NAME="Generator" CONTENT="LaTeX2HTML v2017.2">
|
||||
<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
|
||||
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
||||
|
||||
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
||||
@@ -54,27 +54,27 @@ 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
||||
algorithm described in [<A
|
||||
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">25</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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="43" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img9.png"
|
||||
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="43" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"></SPAN>;
|
||||
</LI>
|
||||
<LI>application of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img15.png"
|
||||
ALT="$A^{k+1}$"></SPAN>.
|
||||
</LI>
|
||||
@@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space
|
||||
<P>
|
||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
||||
described in [<A
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] is used. In this algorithm,
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] is used. In this algorithm,
|
||||
each index <!-- MATH
|
||||
$j \in \Omega^{k+1}$
|
||||
-->
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="71" 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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" 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="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img30.png"
|
||||
ALT="$i$"></SPAN>, i.e.,
|
||||
</BIG></BIG></BIG>
|
||||
@@ -149,11 +149,13 @@ 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="72" BORDER="0"
|
||||
WIDTH="387" HEIGHT="49" BORDER="0"
|
||||
SRC="img31.png"
|
||||
ALT="\begin{displaymath}
|
||||
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
\left\{ r ...
|
||||
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
\end{displaymath}"></TD>
|
||||
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
||||
\left\{ r \i...
|
||||
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},
|
||||
\end{displaymath}"></TD>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
@@ -162,10 +164,10 @@ for a given threshold <!-- MATH
|
||||
$\theta \in [0,1]$
|
||||
-->
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img32.png"
|
||||
ALT="$\theta \in [0,1]$"></SPAN> (see [<A
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">25</A>] for the details).
|
||||
HREF="node36.html#VANEK_MANDEL_BREZINA">26</A>] for the details).
|
||||
Since this algorithm has a sequential nature, a decoupled
|
||||
version of it is applied, where each processor independently executes
|
||||
the algorithm on the set of indices assigned to it in the initial data
|
||||
@@ -180,11 +182,11 @@ MLD2P4, since it has been shown to produce good results in practice
|
||||
[<A
|
||||
HREF="node36.html#aaecc_07">5</A>,<A
|
||||
HREF="node36.html#apnum_07">7</A>,<A
|
||||
HREF="node36.html#TUMINARO_TONG">24</A>].
|
||||
HREF="node36.html#TUMINARO_TONG">25</A>].
|
||||
</BIG></BIG></BIG>
|
||||
<P>
|
||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
||||
<!-- MATH
|
||||
@@ -210,10 +212,14 @@ 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="287" HEIGHT="51" BORDER="0"
|
||||
WIDTH="286" HEIGHT="52" BORDER="0"
|
||||
SRC="img34.png"
|
||||
ALT="\begin{displaymath}
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
\left\{...
|
||||
...ega^k_j,
0 & \quad \mbox{otherwise},
\end{array} \right.
|
||||
ALT="\begin{displaymath}
|
||||
\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k =
|
||||
\left\{...
|
||||
...Omega^k_j, \\
|
||||
0 & \quad \mbox{otherwise},
|
||||
\end{array} \right.
|
||||
\end{displaymath}"></TD>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
||||
@@ -223,21 +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="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" 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="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img27.png"
|
||||
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img25.png"
|
||||
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img35.png"
|
||||
ALT="$\bar{P}^k$"></SPAN> a smoother
|
||||
<!-- MATH
|
||||
@@ -257,9 +263,11 @@ P^k = S^k \bar{P}^k,
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="90" HEIGHT="30" BORDER="0"
|
||||
WIDTH="91" HEIGHT="30" BORDER="0"
|
||||
SRC="img37.png"
|
||||
ALT="\begin{displaymath}
P^k = S^k \bar{P}^k,
\end{displaymath}">
|
||||
ALT="\begin{displaymath}
|
||||
P^k = S^k \bar{P}^k,
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||
@@ -267,9 +275,9 @@ in order to remove nonsmooth components from the range of the prolongator,
|
||||
and hence to improve the convergence properties of the multilevel
|
||||
method [<A
|
||||
HREF="node36.html#BREZINA_VANEK">2</A>,<A
|
||||
HREF="node36.html#Stuben_01">23</A>].
|
||||
HREF="node36.html#Stuben_01">24</A>].
|
||||
A simple choice for <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img38.png"
|
||||
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
||||
</BIG></BIG></BIG>
|
||||
@@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="175" HEIGHT="31" BORDER="0"
|
||||
WIDTH="176" HEIGHT="32" BORDER="0"
|
||||
SRC="img39.png"
|
||||
ALT="\begin{displaymath}
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
\end{displaymath}">
|
||||
ALT="\begin{displaymath}
|
||||
S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||
where <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="28" HEIGHT="19" 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="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"></SPAN>,
|
||||
<!-- MATH
|
||||
$A^k_F = (\bar{a}_{ij}^k)$
|
||||
-->
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="86" 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>
|
||||
@@ -321,17 +331,20 @@ where <SPAN CLASS="MATH"><IMG
|
||||
<A NAME="eq:filtered"></A>
|
||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
||||
WIDTH="514" HEIGHT="74" BORDER="0"
|
||||
WIDTH="499" HEIGHT="59" 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>
|
||||
\bar{a}_{ij}^k =
|
||||
\left \{ \begin{array}{ll}
|
||||
a_{ij}^k & \m...
|
||||
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),
|
||||
\end{displaymath}"></TD>
|
||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
||||
</TABLE>
|
||||
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||
and <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="24" HEIGHT="20" 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"
|
||||
@@ -360,14 +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="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"></SPAN> has little or no effect, and
|
||||
<SPAN CLASS="MATH"><IMG
|
||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
||||
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img41.png"
|
||||
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
|
||||
WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
WIDTH="28" 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>
|
||||
|
||||