mirror of
https://github.com/sfilippone/amg4psblas.git
synced 2026-10-06 22:55:12 +00:00
HTML doc fixes.
This commit is contained in:
Salvatore Filippone
+83
-20
@@ -18,6 +18,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
|
||||
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
||||
|
||||
<LINK REL="next" HREF="node13.html">
|
||||
<LINK REL="previous" HREF="node11.html">
|
||||
<LINK REL="up" HREF="node11.html">
|
||||
<LINK REL="next" HREF="node13.html">
|
||||
@@ -25,26 +26,26 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
||||
|
||||
<BODY >
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html220"
|
||||
<A NAME="tex2html229"
|
||||
HREF="node13.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
||||
<A NAME="tex2html216"
|
||||
<A NAME="tex2html225"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
||||
<A NAME="tex2html212"
|
||||
<A NAME="tex2html219"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
||||
<A NAME="tex2html218"
|
||||
<A NAME="tex2html227"
|
||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html221"
|
||||
HREF="node13.html">Getting Started</A>
|
||||
<B> Up:</B> <A NAME="tex2html217"
|
||||
<B> Next:</B> <A NAME="tex2html230"
|
||||
HREF="node13.html">Smoothed Aggregation</A>
|
||||
<B> Up:</B> <A NAME="tex2html226"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html213"
|
||||
<B> Previous:</B> <A NAME="tex2html220"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> <A NAME="tex2html219"
|
||||
<B> <A NAME="tex2html228"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<BR>
|
||||
<BR>
|
||||
@@ -150,29 +151,91 @@ 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">
|
||||
</FONT></FONT></FONT><HR>
|
||||
<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>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
A^{k+1}=R^kA^kP^k.
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="131" HEIGHT="27" BORDER="0"
|
||||
SRC="img16.png"
|
||||
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
|
||||
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img17.png"
|
||||
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="$k < nlev$">, 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
|
||||
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>
|
||||
<DIV ALIGN="CENTER"><A NAME="fig:application_alg"></A><A NAME="517"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
|
||||
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}␍}$
|
||||
-->
|
||||
<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}␍}">
|
||||
</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
|
||||
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 <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 [<A
|
||||
HREF="node29.html#Briggs2000">3</A>], and a version of the K-cycle described␍in [<A
|
||||
HREF="node29.html#Notay2008">19</A>]. ␍␍␍</FONT></FONT></FONT><HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html220"
|
||||
<A NAME="tex2html229"
|
||||
HREF="node13.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
||||
<A NAME="tex2html216"
|
||||
<A NAME="tex2html225"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
||||
<A NAME="tex2html212"
|
||||
<A NAME="tex2html219"
|
||||
HREF="node11.html">
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
||||
<A NAME="tex2html218"
|
||||
<A NAME="tex2html227"
|
||||
HREF="node2.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html221"
|
||||
HREF="node13.html">Getting Started</A>
|
||||
<B> Up:</B> <A NAME="tex2html217"
|
||||
<B> Next:</B> <A NAME="tex2html230"
|
||||
HREF="node13.html">Smoothed Aggregation</A>
|
||||
<B> Up:</B> <A NAME="tex2html226"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> Previous:</B> <A NAME="tex2html213"
|
||||
<B> Previous:</B> <A NAME="tex2html220"
|
||||
HREF="node11.html">Multigrid Background</A>
|
||||
<B> <A NAME="tex2html219"
|
||||
<B> <A NAME="tex2html228"
|
||||
HREF="node2.html">Contents</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
|
||||
|
||||
Reference in New Issue
Block a user