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<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>
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
Ax=b,
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:system"></A><IMG
WIDTH="58" HEIGHT="30" BORDER="0"
SRC="img2.png"
ALT="\begin{displaymath}
Ax=b,
\end{displaymath}"></TD>
<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
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
<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
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
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
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>
<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},
\end{displaymath}
-->
<IMG
WIDTH="398" HEIGHT="30" BORDER="0"
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}">
</DIV>
<BR CLEAR="ALL">
<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
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$A$"> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
<IMG
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
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
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
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
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>
<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};
\end{displaymath}
-->
<IMG
WIDTH="254" 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}">
</DIV>
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