fdurastante/amg4psblas
1
0
Fork 0
mirror of https://github.com/sfilippone/amg4psblas.git synced 2026-10-07 07:04:59 +00:00
Code Issues Packages Projects Releases Wiki Activity

Fixed docs.

Browse Source
This commit is contained in:
Salvatore Filippone
2018-05-14 14:50:52 +01:00
parent 4b43164668
commit b7e8a921d8
101 changed files with 85 additions and 85 deletions
Download Patch File Download Diff File Expand all files Collapse all files
docs/html/node13.html
+23 -23
View File
@@ -80,25 +80,25 @@ where <!-- MATH
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img5.png"
ALT="$A=(a_{ij}) \in \mathbb{R}^{n \times n}$"></SPAN> is a nonsingular sparse matrix;
for ease of presentation we assume <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> has a symmetric sparsity
pattern.
</BIG></BIG></BIG>
<P>
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Let us consider as finest index space the set of row (column) indices of <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>, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img6.png"
ALT="$\Omega = \{1, 2, \ldots, n\}$"></SPAN>.
Any algebraic multilevel preconditioners implemented in MLD2P4 generates
@@ -122,39 +122,39 @@ a hierarchy of index spaces and a corresponding hierarchy of matrices,
<BR CLEAR="ALL">
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
by using the information contained in <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>, without assuming any
knowledge of the geometry of the problem from which <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> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<SPAN CLASS="MATH"><IMG
WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="34" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\mathbb{R}^{n_{k}}$"></SPAN> is associated with <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>,
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="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img9.png"
ALT="$\Omega^k$"></SPAN>.
For all <SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$"></SPAN>, a restriction operator and a prolongation one are built,
which connect two levels <SPAN CLASS="MATH"><IMG
WIDTH="14" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img12.png"
ALT="$k$"></SPAN> and <SPAN CLASS="MATH"><IMG
WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img13.png"
ALT="$k+1$"></SPAN>:
</BIG></BIG></BIG>
@@ -168,7 +168,7 @@ P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
-->
<IMG
WIDTH="255" HEIGHT="30" BORDER="0"
WIDTH="253" HEIGHT="30" BORDER="0"
SRC="img14.png"
ALT="\begin{displaymath}
P^k \in \mathbb{R}^{n_k \times n_{k+1}}, \quad
@@ -178,7 +178,7 @@ R^k \in \mathbb{R}^{n_{k+1}\times n_k};
<BR CLEAR="ALL">
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
the 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> is computed by using the previous operators according
to the Galerkin approach, i.e.,
@@ -192,7 +192,7 @@ A^{k+1}=R^kA^kP^k.
-->
<IMG
WIDTH="131" HEIGHT="28" BORDER="0"
WIDTH="129" HEIGHT="27" BORDER="0"
SRC="img16.png"
ALT="\begin{displaymath}
A^{k+1}=R^kA^kP^k.
@@ -205,22 +205,22 @@ 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="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img18.png"
ALT="$M^k$"></SPAN> is set up at each level <SPAN CLASS="MATH"><IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img11.png"
ALT="$k &lt; nlev$"></SPAN>, 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 <SPAN CLASS="MATH"><IMG
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
WIDTH="30" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img19.png"
ALT="$LU$"></SPAN> factorization means computing
and storing the <SPAN CLASS="MATH"><IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
WIDTH="17" 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"
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$U$"></SPAN> factors). The construction of the hierarchy of AMG components
described so far corresponds to the so-called build phase of the preconditioner.
@@ -262,8 +262,8 @@ end
\begin{tabbing}
\quad \=\quad \=\quad...
...[1mm]
\&gt;endif [1mm]
\&gt;return $u^k$ [1mm]
\&gt;endif \\ [1mm]
\&gt;return $u^k$\ \\ [1mm]
end
\end{tabbing}\end{minipage}}">
@@ -276,7 +276,7 @@ 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="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
WIDTH="82" HEIGHT="21" 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"