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< H2 > < A NAME = "SECTION00061000000000000000" > < / A > < A NAME = "sec:multilevel" > < / A >
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AMG preconditioners
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In order to describe the AMG preconditioners available in MLD2P4, we consider a
linear system
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<!-- 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="img1.png"
ALT="\begin{displaymath}
Ax=b,
\end{displaymath}">< / TD >
< TD WIDTH = 10 ALIGN = "RIGHT" >
(2)< / TD > < / TR >
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< BR CLEAR = "ALL" > < / DIV > < P > < / P >
where <!-- MATH
$A=(a_{ij}) \in \mathbb{R}^{n \times n}$
-->
< IMG
WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
SRC="img4.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="img2.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="img2.png"
ALT="$A$">, i.e.,
<!-- MATH
$\Omega = \{1, 2, \ldots, n\}$
-->
< IMG
WIDTH="132" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img5.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,
< 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="img6.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 >
by using the information contained in < IMG
WIDTH="18" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img2.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="img2.png"
ALT="$A$"> originates.
A vector space <!-- MATH
$\mathbb{R}^{n_{k}}$
-->
< IMG
WIDTH="33" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img7.png"
ALT="$\mathbb{R}^{n_{k}}$"> is associated with < IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\Omega^k$">,
where < IMG
WIDTH="23" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img9.png"
ALT="$n_k$"> is the size of < IMG
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img8.png"
ALT="$\Omega^k$">.
For all < IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img10.png"
ALT="$k < nlev$">, a restriction operator and a prolongation one are built,
which connect two levels < IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img11.png"
ALT="$k$"> and < IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img12.png"
ALT="$k+1$">:
< BR > < P > < / P >
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<!-- 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="img13.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}">
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