Fixup the documentation.
Salvatore Filippone
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Software version: 2.1
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Software version: 2.2
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July 31, 2017
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July 31, 2018
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<TITLE>Abstract</TITLE>
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<TITLE>Abstract</TITLE>
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<TITLE>Bug reporting</TITLE>
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<TITLE>Example and test programs</TITLE>
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<TITLE>Multigrid Background</TITLE>
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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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interplay between fine and coarse level by suitably choosing the coarse space and
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the coarse-to-fine interpolation (see, e.g., [<A
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the coarse-to-fine interpolation (see, e.g., [<A
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HREF="node36.html#Briggs2000">3</A>,<A
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HREF="node36.html#Briggs2000">3</A>,<A
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HREF="node36.html#Stuben_01">23</A>,<A
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HREF="node36.html#Stuben_01">24</A>,<A
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HREF="node36.html#dd2_96">21</A>] for details.)
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HREF="node36.html#dd2_96">22</A>] for details.)
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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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<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
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aggregation algorithm [<A
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HREF="node36.html#BREZINA_VANEK">2</A>,<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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for building the sequence of coarse matrices and transfer operators,
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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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starting from the original one.
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A decoupled version of this algorithm is implemented, where the smoothed
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A decoupled version of this algorithm is implemented, where the smoothed
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aggregation is applied locally to each submatrix [<A
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aggregation is applied locally to each submatrix [<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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A brief description of the AMG preconditioners implemented in MLD2P4 is given in
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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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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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is referred to [<A
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<TITLE>AMG preconditioners</TITLE>
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<TITLE>AMG preconditioners</TITLE>
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<A NAME="eq:system"></A>
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<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
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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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$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
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-->
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-->
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<SPAN CLASS="MATH"><IMG
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SRC="img5.png"
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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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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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for ease of presentation we assume <SPAN CLASS="MATH"><IMG
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$\Omega = \{1, 2, \ldots, n\}$
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$\Omega = \{1, 2, \ldots, n\}$
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-->
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ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>.
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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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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,
|
|||||||
<IMG
|
<IMG
|
||||||
WIDTH="398" HEIGHT="30" BORDER="0"
|
WIDTH="398" HEIGHT="30" BORDER="0"
|
||||||
SRC="img7.png"
|
SRC="img7.png"
|
||||||
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>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
@@ -132,28 +133,28 @@ A vector space <!-- MATH
|
|||||||
$\mathbb{R}^{n_{k}}$
|
$\mathbb{R}^{n_{k}}$
|
||||||
-->
|
-->
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img8.png"
|
SRC="img8.png"
|
||||||
ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN>,
|
ALT="$\Omega^k$"></SPAN>,
|
||||||
where <SPAN CLASS="MATH"><IMG
|
where <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img10.png"
|
SRC="img10.png"
|
||||||
ALT="$n_k$"></SPAN> is the size of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN>.
|
ALT="$\Omega^k$"></SPAN>.
|
||||||
For all <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img11.png"
|
||||||
ALT="$k < nlev$"></SPAN>, a restriction operator and a prolongation one are built,
|
ALT="$k < nlev$"></SPAN>, a restriction operator and a prolongation one are built,
|
||||||
which connect two levels <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img12.png"
|
||||||
ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img13.png"
|
||||||
ALT="$k+1$"></SPAN>:
|
ALT="$k+1$"></SPAN>:
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
@@ -167,14 +168,17 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
|
|||||||
-->
|
-->
|
||||||
|
|
||||||
<IMG
|
<IMG
|
||||||
WIDTH="254" HEIGHT="30" BORDER="0"
|
WIDTH="255" HEIGHT="30" BORDER="0"
|
||||||
SRC="img14.png"
|
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>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
the matrix <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img15.png"
|
||||||
ALT="$A^{k+1}$"></SPAN> is computed by using the previous operators according
|
ALT="$A^{k+1}$"></SPAN> is computed by using the previous operators according
|
||||||
to the Galerkin approach, i.e.,
|
to the Galerkin approach, i.e.,
|
||||||
@@ -188,9 +192,11 @@ A^{k+1}=R^kA^kP^k.
|
|||||||
-->
|
-->
|
||||||
|
|
||||||
<IMG
|
<IMG
|
||||||
WIDTH="131" HEIGHT="27" BORDER="0"
|
WIDTH="131" HEIGHT="28" BORDER="0"
|
||||||
SRC="img16.png"
|
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>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<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"
|
SRC="img17.png"
|
||||||
ALT="$R^k=(P^k)^T$"></SPAN>
|
ALT="$R^k=(P^k)^T$"></SPAN>
|
||||||
A smoother with iteration matrix <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img18.png"
|
||||||
ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img11.png"
|
||||||
ALT="$k < nlev$"></SPAN>, and a solver
|
ALT="$k < nlev$"></SPAN>, and a solver
|
||||||
is set up at the coarsest level, so that they are ready for application
|
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"
|
SRC="img19.png"
|
||||||
ALT="$LU$"></SPAN> factorization means computing
|
ALT="$LU$"></SPAN> factorization means computing
|
||||||
and storing the <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img20.png"
|
||||||
ALT="$L$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
ALT="$L$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
|
||||||
@@ -251,8 +257,15 @@ end
|
|||||||
<IMG
|
<IMG
|
||||||
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img22.png"
|
SRC="img22.png"
|
||||||
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
|
ALT="\framebox{
|
||||||
...mm]␍\>endif [1mm]␍\>return $u^k$ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
|
\begin{minipage}{.85\textwidth}
|
||||||
|
\begin{tabbing}
|
||||||
|
\quad \=\quad \=\quad...
|
||||||
|
...[1mm]
|
||||||
|
\>endif [1mm]
|
||||||
|
\>return $u^k$ [1mm]
|
||||||
|
end
|
||||||
|
\end{tabbing}\end{minipage}}">
|
||||||
|
|
||||||
</DIV></TD></TR>
|
</DIV></TD></TR>
|
||||||
</TABLE>
|
</TABLE>
|
||||||
@@ -263,14 +276,14 @@ end
|
|||||||
to obtain different multilevel preconditioners;
|
to obtain different multilevel preconditioners;
|
||||||
this is done in the application phase, i.e., in the computation of a vector
|
this is done in the application phase, i.e., in the computation of a vector
|
||||||
of type <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img23.png"
|
||||||
ALT="$w=B^{-1}v$"></SPAN>, where <SPAN CLASS="MATH"><IMG
|
ALT="$w=B^{-1}v$"></SPAN>, where <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img24.png"
|
SRC="img24.png"
|
||||||
ALT="$B$"></SPAN> denotes the preconditioner, usually within an iteration
|
ALT="$B$"></SPAN> denotes the preconditioner, usually within an iteration
|
||||||
of a Krylov solver [<A
|
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
|
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.,
|
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
|
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
|
and W-cycle [<A
|
||||||
HREF="node36.html#Briggs2000">3</A>], and a version of the K-cycle described
|
HREF="node36.html#Briggs2000">3</A>], and a version of the K-cycle described
|
||||||
in [<A
|
in [<A
|
||||||
HREF="node36.html#Notay2008">19</A>].
|
HREF="node36.html#Notay2008">20</A>].
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
<P>
|
<P>
|
||||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
|
|||||||
@@ -1,6 +1,6 @@
|
|||||||
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
<!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>
|
<HTML>
|
||||||
<HEAD>
|
<HEAD>
|
||||||
<TITLE>Smoothed Aggregation</TITLE>
|
<TITLE>Smoothed Aggregation</TITLE>
|
||||||
@@ -9,7 +9,7 @@
|
|||||||
<META NAME="resource-type" CONTENT="document">
|
<META NAME="resource-type" CONTENT="document">
|
||||||
<META NAME="distribution" CONTENT="global">
|
<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">
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
||||||
|
|
||||||
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
||||||
@@ -54,27 +54,27 @@ Smoothed Aggregation
|
|||||||
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
||||||
<P>
|
<P>
|
||||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to define the prolongator <SPAN CLASS="MATH"><IMG
|
<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"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN>, used to compute
|
ALT="$P^k$"></SPAN>, used to compute
|
||||||
the coarse-level matrix <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img15.png"
|
||||||
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
ALT="$A^{k+1}$"></SPAN>, MLD2P4 uses the smoothed aggregation
|
||||||
algorithm described in [<A
|
algorithm described in [<A
|
||||||
HREF="node36.html#BREZINA_VANEK">2</A>,<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
|
The basic idea of this algorithm is to build a coarse set of indices
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img26.png"
|
SRC="img26.png"
|
||||||
ALT="$\Omega^{k+1}$"></SPAN> by suitably grouping the indices of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN> into disjoint
|
ALT="$\Omega^k$"></SPAN> into disjoint
|
||||||
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
subsets (aggregates), and to define the coarse-to-fine space transfer operator
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img25.png"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN> by applying a suitable smoother to a simple piecewise constant
|
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.
|
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>
|
</BIG></BIG></BIG>
|
||||||
<OL>
|
<OL>
|
||||||
<LI>aggregation of the indices of <SPAN CLASS="MATH"><IMG
|
<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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN> to obtain <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img26.png"
|
||||||
ALT="$\Omega^{k+1}$"></SPAN>;
|
ALT="$\Omega^{k+1}$"></SPAN>;
|
||||||
</LI>
|
</LI>
|
||||||
<LI>construction of the prolongator <SPAN CLASS="MATH"><IMG
|
<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"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN>;
|
ALT="$P^k$"></SPAN>;
|
||||||
</LI>
|
</LI>
|
||||||
<LI>application of <SPAN CLASS="MATH"><IMG
|
<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"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
ALT="$P^k$"></SPAN> and <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img17.png"
|
SRC="img17.png"
|
||||||
ALT="$R^k=(P^k)^T$"></SPAN> to build <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img15.png"
|
||||||
ALT="$A^{k+1}$"></SPAN>.
|
ALT="$A^{k+1}$"></SPAN>.
|
||||||
</LI>
|
</LI>
|
||||||
@@ -111,18 +111,18 @@ prolongation operator, with the aim of improving the quality of the coarse-space
|
|||||||
<P>
|
<P>
|
||||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In order to perform the coarsening step, the smoothed aggregation algorithm
|
||||||
described in [<A
|
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
|
each index <!-- MATH
|
||||||
$j \in \Omega^{k+1}$
|
$j \in \Omega^{k+1}$
|
||||||
-->
|
-->
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img27.png"
|
SRC="img27.png"
|
||||||
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
|
ALT="$j \in \Omega^{k+1}$"></SPAN> corresponds to an aggregate <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img28.png"
|
SRC="img28.png"
|
||||||
ALT="$\Omega^k_j$"></SPAN> of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN>,
|
ALT="$\Omega^k$"></SPAN>,
|
||||||
consisting of a suitably chosen index <!-- MATH
|
consisting of a suitably chosen index <!-- MATH
|
||||||
@@ -133,7 +133,7 @@ consisting of a suitably chosen index <!-- MATH
|
|||||||
SRC="img29.png"
|
SRC="img29.png"
|
||||||
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
|
ALT="$i \in \Omega^k$"></SPAN> and indices that are (usually) contained in a
|
||||||
strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img30.png"
|
||||||
ALT="$i$"></SPAN>, i.e.,
|
ALT="$i$"></SPAN>, i.e.,
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
@@ -149,11 +149,13 @@ strongly-coupled neighborood of <SPAN CLASS="MATH"><IMG
|
|||||||
<A NAME="eq:strongly_coup"></A>
|
<A NAME="eq:strongly_coup"></A>
|
||||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:strongly_coup"></A><IMG
|
<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"
|
SRC="img31.png"
|
||||||
ALT="\begin{displaymath}
|
ALT="\begin{displaymath}
|
||||||
\Omega^k_j \subset \mathcal{N}_i^k(\theta) = ␍ \left\{ r ...
|
\Omega^k_j \subset \mathcal{N}_i^k(\theta) =
|
||||||
...vert a_{ii}^ka_{rr}^k\vert} \right \} \cup \left\{ i \right\},␍\end{displaymath}"></TD>
|
\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">
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||||
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
(<SPAN CLASS="arabic">3</SPAN>)</TD></TR>
|
||||||
</TABLE>
|
</TABLE>
|
||||||
@@ -162,10 +164,10 @@ for a given threshold <!-- MATH
|
|||||||
$\theta \in [0,1]$
|
$\theta \in [0,1]$
|
||||||
-->
|
-->
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="69" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img32.png"
|
SRC="img32.png"
|
||||||
ALT="$\theta \in [0,1]$"></SPAN> (see [<A
|
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
|
Since this algorithm has a sequential nature, a decoupled
|
||||||
version of it is applied, where each processor independently executes
|
version of it is applied, where each processor independently executes
|
||||||
the algorithm on the set of indices assigned to it in the initial data
|
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
|
[<A
|
||||||
HREF="node36.html#aaecc_07">5</A>,<A
|
HREF="node36.html#aaecc_07">5</A>,<A
|
||||||
HREF="node36.html#apnum_07">7</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>
|
</BIG></BIG></BIG>
|
||||||
<P>
|
<P>
|
||||||
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The prolongator <SPAN CLASS="MATH"><IMG
|
<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"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
ALT="$P^k$"></SPAN> is built starting from a tentative prolongator
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -210,10 +212,14 @@ MLD2P4, since it has been shown to produce good results in practice
|
|||||||
<A NAME="eq:tent_prol"></A>
|
<A NAME="eq:tent_prol"></A>
|
||||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:tent_prol"></A><IMG
|
<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"
|
SRC="img34.png"
|
||||||
ALT="\begin{displaymath}␍\bar{P}^k =(\bar{p}_{ij}^k), \quad \bar{p}_{ij}^k = ␍\left\{...
|
ALT="\begin{displaymath}
|
||||||
...ega^k_j, ␍0 & \quad \mbox{otherwise},␍\end{array} \right.
|
\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>
|
\end{displaymath}"></TD>
|
||||||
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||||
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
(<SPAN CLASS="arabic">4</SPAN>)</TD></TR>
|
||||||
@@ -223,21 +229,21 @@ where <SPAN CLASS="MATH"><IMG
|
|||||||
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img28.png"
|
SRC="img28.png"
|
||||||
ALT="$\Omega^k_j$"></SPAN> is the aggregate of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img9.png"
|
||||||
ALT="$\Omega^k$"></SPAN>
|
ALT="$\Omega^k$"></SPAN>
|
||||||
corresponding to the index <!-- MATH
|
corresponding to the index <!-- MATH
|
||||||
$j \in \Omega^{k+1}$
|
$j \in \Omega^{k+1}$
|
||||||
-->
|
-->
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img27.png"
|
SRC="img27.png"
|
||||||
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
ALT="$j \in \Omega^{k+1}$"></SPAN>.
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img25.png"
|
SRC="img25.png"
|
||||||
ALT="$P^k$"></SPAN> is obtained by applying to <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img35.png"
|
||||||
ALT="$\bar{P}^k$"></SPAN> a smoother
|
ALT="$\bar{P}^k$"></SPAN> a smoother
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -257,9 +263,11 @@ P^k = S^k \bar{P}^k,
|
|||||||
-->
|
-->
|
||||||
|
|
||||||
<IMG
|
<IMG
|
||||||
WIDTH="90" HEIGHT="30" BORDER="0"
|
WIDTH="91" HEIGHT="30" BORDER="0"
|
||||||
SRC="img37.png"
|
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>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<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
|
and hence to improve the convergence properties of the multilevel
|
||||||
method [<A
|
method [<A
|
||||||
HREF="node36.html#BREZINA_VANEK">2</A>,<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
|
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"
|
SRC="img38.png"
|
||||||
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
ALT="$S^k$"></SPAN> is the damped Jacobi smoother:
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
@@ -282,24 +290,26 @@ S^k = I - \omega^k (D^k)^{-1} A^k_F ,
|
|||||||
-->
|
-->
|
||||||
|
|
||||||
<IMG
|
<IMG
|
||||||
WIDTH="175" HEIGHT="31" BORDER="0"
|
WIDTH="176" HEIGHT="32" BORDER="0"
|
||||||
SRC="img39.png"
|
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>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
where <SPAN CLASS="MATH"><IMG
|
where <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img40.png"
|
SRC="img40.png"
|
||||||
ALT="$D^k$"></SPAN> is the diagonal matrix with the same diagonal entries as <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img41.png"
|
||||||
ALT="$A^k$"></SPAN>,
|
ALT="$A^k$"></SPAN>,
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
$A^k_F = (\bar{a}_{ij}^k)$
|
$A^k_F = (\bar{a}_{ij}^k)$
|
||||||
-->
|
-->
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img42.png"
|
SRC="img42.png"
|
||||||
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
|
ALT="$A^k_F = (\bar{a}_{ij}^k)$"></SPAN> is the filtered matrix defined as
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
@@ -321,17 +331,20 @@ where <SPAN CLASS="MATH"><IMG
|
|||||||
<A NAME="eq:filtered"></A>
|
<A NAME="eq:filtered"></A>
|
||||||
<TABLE WIDTH="100%" ALIGN="CENTER">
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
||||||
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:filtered"></A><IMG
|
<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"
|
SRC="img43.png"
|
||||||
ALT="\begin{displaymath}
|
ALT="\begin{displaymath}
|
||||||
\bar{a}_{ij}^k =␍ \left \{ \begin{array}{ll}␍ a_{ij}^k & ...
|
\bar{a}_{ij}^k =
|
||||||
...ii}^k = a_{ii}^k - \sum_{j \ne i} (a_{ij}^k - \bar{a}_{ij}^k),␍\end{displaymath}"></TD>
|
\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">
|
<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT">
|
||||||
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
(<SPAN CLASS="arabic">5</SPAN>)</TD></TR>
|
||||||
</TABLE>
|
</TABLE>
|
||||||
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
<BR CLEAR="ALL"></DIV><P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
||||||
and <SPAN CLASS="MATH"><IMG
|
and <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img44.png"
|
SRC="img44.png"
|
||||||
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
|
ALT="$\omega^k$"></SPAN> is an approximation of <SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
@@ -360,14 +373,14 @@ of <SPAN CLASS="MATH"><IMG
|
|||||||
SRC="img46.png"
|
SRC="img46.png"
|
||||||
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
|
ALT="$\rho^k$"></SPAN>. Note that for systems coming from uniformly elliptic
|
||||||
problems, filtering the matrix <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img41.png"
|
||||||
ALT="$A^k$"></SPAN> has little or no effect, and
|
ALT="$A^k$"></SPAN> has little or no effect, and
|
||||||
<SPAN CLASS="MATH"><IMG
|
<SPAN CLASS="MATH"><IMG
|
||||||
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="25" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img41.png"
|
SRC="img41.png"
|
||||||
ALT="$A^k$"></SPAN> can be used instead of <SPAN CLASS="MATH"><IMG
|
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"
|
SRC="img49.png"
|
||||||
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
|
ALT="$A^k_F$"></SPAN>. The latter choice is the default in MLD2P4.
|
||||||
</BIG></BIG></BIG>
|
</BIG></BIG></BIG>
|
||||||
|
|||||||