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Fix from Mac coding to Unix coding.

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Salvatore Filippone
2017-07-25 14:19:05 +01:00
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@@ -54,7 +54,11 @@ original version by: Nikos Drakos, CBLU, University of Leeds
<H2><A NAME="SECTION00061000000000000000"></A><A NAME="sec:multilevel"></A>
<BR>
AMG preconditioners
</H2><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍In order to describe the AMG preconditioners available in MLD2P4, we consider a␍linear system␍</FONT></FONT></FONT>
</H2><FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
</FONT></FONT></FONT>
<BR>
<DIV ALIGN="RIGHT">
@@ -73,30 +77,41 @@ Ax=b,
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍where <!-- MATH
<BR CLEAR="ALL"></DIV><P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
where <!-- MATH
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
<IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;␍for ease of presentation we assume <IMG
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"> is a nonsingular sparse matrix;
for ease of presentation we assume <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> is real, but the␍results are valid for the complex case as well. ␍␍Let us assume as finest index space the set of row (column) indices of <IMG
ALT="$A$"> is real, but the
results are valid for the complex case as well.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">Let us assume as finest index space the set of row (column) indices of <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, i.e.,␍<!-- MATH
ALT="$A$">, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$\Omega = \{1, 2, \ldots, n\}$">. ␍Any algebraic multilevel preconditioners implemented in MLD2P4 generates␍a hierarchy of index spaces and a corresponding hierarchy of matrices,␍</FONT></FONT></FONT>
ALT="$\Omega = \{1, 2, \ldots, n\}$">.
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
a hierarchy of index spaces and a corresponding hierarchy of matrices,
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},␍\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
\Omega^1 \equiv \Omega \supset \Omega^2 \supset \ldots \supset \Omega^{nlev},
\quad A^1 \equiv A, A^2, \ldots, A^{nlev},
\end{displaymath}
-->
@@ -106,13 +121,16 @@ Ax=b,
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>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍by using the information contained in <IMG
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
by using the information contained in <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$">, without assuming any␍knowledge of the geometry of the problem from which <IMG
ALT="$A$">, without assuming any
knowledge of the geometry of the problem from which <IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> originates.␍A vector space <!-- MATH
ALT="$A$"> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<IMG
@@ -121,27 +139,32 @@ Ax=b,
ALT="$\mathbb{R}^{n_{k}}$"> is associated with <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">,␍where <IMG
ALT="$\Omega^k$">,
where <IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
ALT="$n_k$"> is the size of <IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$">.␍For all <IMG
ALT="$\Omega^k$">.
For all <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$">, a restriction operator and a prolongation one are built,␍which connect two levels <IMG
ALT="$k &lt; nlev$">, a restriction operator and a prolongation one are built,
which connect two levels <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$k$"> and <IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
ALT="$k+1$">:␍</FONT></FONT></FONT>
ALT="$k+1$">:
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\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};
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}
-->
@@ -151,10 +174,13 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad R^k \in \mathbb{R}^{n_{k+1}\
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><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍the matrix <IMG
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
the matrix <IMG
WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img15.png"
ALT="$A^{k+1}$"> is computed by using the previous operators according␍to the Galerkin approach, i.e.,␍</FONT></FONT></FONT>
ALT="$A^{k+1}$"> is computed by using the previous operators according
to the Galerkin approach, i.e.,
</FONT></FONT></FONT>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
@@ -169,25 +195,34 @@ A^{k+1}=R^kA^kP^k.
ALT="\begin{displaymath}␍ A^{k+1}=R^kA^kP^k.␍\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍In the current implementation of MLD2P4 we have <IMG
<P></P><FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">
In the current implementation of MLD2P4 we have <IMG
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img17.png"
ALT="$R^k=(P^k)^T$">␍A smoother with iteration matrix <IMG
ALT="$R^k=(P^k)^T$">
A smoother with iteration matrix <IMG
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img18.png"
ALT="$M^k$"> is set up at each level <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$">, and a solver␍is set up at the coarsest level, so that they are ready for application ␍(for example, setting up a solver based on the <IMG
ALT="$k &lt; nlev$">, and a solver
is set up at the coarsest level, so that they are ready for application
(for example, setting up a solver based on the <IMG
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img19.png"
ALT="$LU$"> factorization means computing␍and storing the <IMG
ALT="$LU$"> factorization means computing
and storing the <IMG
WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$L$"> and <IMG
WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$U$"> factors). The construction of the hierarchy of AMG components␍described so far corresponds to the so-called build phase of the preconditioner.␍␍</FONT></FONT></FONT>
ALT="$U$"> factors). The construction of the hierarchy of AMG components
described so far corresponds to the so-called build phase of the preconditioner.
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
@@ -195,26 +230,60 @@ Application phase of a V-cycle preconditioner.</CAPTION>
<TR><TD>
<DIV ALIGN="CENTER">
<!-- MATH
$\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad \=\quad \\[-3mm]␍procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]␍\>if $\left(k \ne nlev \right)$\ then \\[1mm]␍\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]␍\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]␍\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]␍\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]␍\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]␍\>else \\[1mm]␍\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]␍\>endif \\[1mm]␍\>return $u^k$\ \\[1mm]␍end␍\end{tabbing}␍\end{minipage}␍}$
$\framebox{
\begin{minipage}{.85\textwidth}
\begin{tabbing}
\quad \=\quad \=\quad \=\quad \\[-3mm]
procedure V-cycle$\left(k,A^k,b^k,u^k\right)$\ \\[2mm]
\>if $\left(k \ne nlev \right)$\ then \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $b^{k+1} = R^{k+1}\left(b^k - A^k u^k\right)$\ \\[1mm]
\>\> $u^{k+1} =$\ V-cycle$\left(k+1,A^{k+1},b^{k+1},0\right)$\ \\[1mm]
\>\> $u^k = u^k + P^{k+1} u^{k+1}$\ \\[1mm]
\>\> $u^k = u^k + M^k \left(b^k - A^k u^k\right)$\ \\[1mm]
\>else \\[1mm]
\>\> $u^k = \left(A^k\right)^{-1} b^k$\\[1mm]
\>endif \\[1mm]
\>return $u^k$\ \\[1mm]
end
\end{tabbing}
\end{minipage}
}$
-->
<IMG
WIDTH="333" HEIGHT="336" ALIGN="BOTTOM" BORDER="0"
SRC="img22.png"
ALT="\framebox{␍\begin{minipage}{.85\textwidth}␍\begin{tabbing}␍\quad \=\quad \=\quad...
...mm]␍\&gt;endif \\ [1mm]␍\&gt;return $u^k$\ \\ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
...mm]␍\&gt;endif \ [1mm]␍\&gt;return $u^k$ \ [1mm]␍end␍\end{tabbing}␍\end{minipage}␍}">
</DIV></TD></TR>
</TABLE>
</DIV>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">␍␍The components produced in the build phase may be combined in several ways␍to obtain different multilevel preconditioners;␍this is done in the application phase, i.e., in the computation of a vector␍of type <IMG
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"><FONT SIZE="+1">The components produced in the build phase may be combined in several ways
to obtain different multilevel preconditioners;
this is done in the application phase, i.e., in the computation of a vector
of type <IMG
WIDTH="82" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$w=B^{-1}v$">, where <IMG
WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img24.png"
ALT="$B$"> denotes the preconditioner, usually within an iteration␍of a Krylov solver [<A
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as␍V-cycle, is given in Figure&nbsp;<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␍performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle␍and W-cycle&nbsp;[<A
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described␍in&nbsp;[<A
HREF="node29.html#Notay2008">19</A>]. ␍␍␍</FONT></FONT></FONT><HR>
ALT="$B$"> denotes the preconditioner, usually within an iteration
of a Krylov solver [<A
HREF="node29.html#Saad_book">20</A>]. An example of such a combination, known as
V-cycle, is given in Figure&nbsp;<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
performed. Other cycles can be defined; in MLD2P4, we implemented the standard V-cycle
and W-cycle&nbsp;[<A
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described
in&nbsp;[<A
HREF="node29.html#Notay2008">19</A>].
</FONT></FONT></FONT>
<P>
<FONT SIZE="+1"><FONT SIZE="+1"></FONT></FONT><HR>
<!--Navigation Panel-->
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HREF="node13.html">