|
|
|
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
|
|
|
|
|
|
|
|
|
|
<!--Converted with LaTeX2HTML 2018 (Released Feb 1, 2018) -->
|
|
|
|
|
<HTML>
|
|
|
|
|
<HEAD>
|
|
|
|
|
<TITLE>Smoothers and coarsest-level solvers</TITLE>
|
|
|
|
|
<META NAME="description" CONTENT="Smoothers and coarsest-level solvers">
|
|
|
|
|
<META NAME="keywords" CONTENT="userhtml">
|
|
|
|
|
<META NAME="resource-type" CONTENT="document">
|
|
|
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
|
|
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2018">
|
|
|
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
|
|
|
|
|
|
<LINK REL="STYLESHEET" HREF="userhtml.css">
|
|
|
|
|
|
|
|
|
|
<LINK REL="previous" HREF="node14.html">
|
|
|
|
|
<LINK REL="up" HREF="node12.html">
|
|
|
|
|
<LINK REL="next" HREF="node16.html">
|
|
|
|
|
</HEAD>
|
|
|
|
|
|
|
|
|
|
<BODY >
|
|
|
|
|
|
|
|
|
|
<DIV CLASS="navigation"><!--Navigation Panel-->
|
|
|
|
|
<A NAME="tex2html271"
|
|
|
|
|
HREF="node16.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
|
|
|
<A NAME="tex2html267"
|
|
|
|
|
HREF="node12.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
|
|
|
<A NAME="tex2html263"
|
|
|
|
|
HREF="node14.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
|
|
|
<A NAME="tex2html269"
|
|
|
|
|
HREF="node2.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
|
|
|
<BR>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html272"
|
|
|
|
|
HREF="node16.html">Getting Started</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html268"
|
|
|
|
|
HREF="node12.html">Multigrid Background</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html264"
|
|
|
|
|
HREF="node14.html">Smoothed Aggregation</A>
|
|
|
|
|
<B> <A NAME="tex2html270"
|
|
|
|
|
HREF="node2.html">Contents</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR></DIV>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
|
|
|
|
<H2><A NAME="SECTION00063000000000000000"></A><A NAME="sec:smoothers"></A>
|
|
|
|
|
<BR>
|
|
|
|
|
Smoothers and coarsest-level solvers
|
|
|
|
|
</H2><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG>
|
|
|
|
|
<P>
|
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The smoothers implemented in MLD2P4 include the Jacobi and block-Jacobi methods,
|
|
|
|
|
a hybrid version of the forward and backward Gauss-Seidel methods, and the
|
|
|
|
|
additive Schwarz (AS) ones (see, e.g., [<A
|
|
|
|
|
HREF="node36.html#Saad_book">21</A>,<A
|
|
|
|
|
HREF="node36.html#dd2_96">22</A>]).
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<P>
|
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">The hybrid Gauss-Seidel
|
|
|
|
|
version is considered because the original Gauss-Seidel method is inherently sequential.
|
|
|
|
|
At each iteration of the hybrid version, each parallel process uses the most recent values
|
|
|
|
|
of its own local variables and the values of the non-local variables computed at the
|
|
|
|
|
previous iteration, obtained by exchanging data with other processes before
|
|
|
|
|
the beginning of the current iteration.
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<P>
|
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">In the AS methods, the index space <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img9.png"
|
|
|
|
|
ALT="$\Omega^k$"></SPAN> is divided into <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="28" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img50.png"
|
|
|
|
|
ALT="$m_k$"></SPAN>
|
|
|
|
|
subsets <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img51.png"
|
|
|
|
|
ALT="$\Omega^k_i$"></SPAN> of size <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img52.png"
|
|
|
|
|
ALT="$n_{k,i}$"></SPAN>, possibly
|
|
|
|
|
overlapping. For each <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img30.png"
|
|
|
|
|
ALT="$i$"></SPAN> we consider the restriction
|
|
|
|
|
operator <!-- MATH
|
|
|
|
|
$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img53.png"
|
|
|
|
|
ALT="$R_i^k \in \mathbb{R}^{n_{k,i} \times n_k}$"></SPAN>
|
|
|
|
|
that maps a vector <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img54.png"
|
|
|
|
|
ALT="$x^k$"></SPAN> to the vector <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="22" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img55.png"
|
|
|
|
|
ALT="$x_i^k$"></SPAN> made of the components of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img54.png"
|
|
|
|
|
ALT="$x^k$"></SPAN>
|
|
|
|
|
with indices in <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img51.png"
|
|
|
|
|
ALT="$\Omega^k_i$"></SPAN>, and the prolongation operator
|
|
|
|
|
<!-- MATH
|
|
|
|
|
$P^k_i = (R_i^k)^T$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img56.png"
|
|
|
|
|
ALT="$P^k_i = (R_i^k)^T$"></SPAN>. These operators are then used to build
|
|
|
|
|
<!-- MATH
|
|
|
|
|
$A_i^k=R_i^kA^kP_i^k$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="113" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img57.png"
|
|
|
|
|
ALT="$A_i^k=R_i^kA^kP_i^k$"></SPAN>, which is the restriction of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img41.png"
|
|
|
|
|
ALT="$A^k$"></SPAN> to the index
|
|
|
|
|
space <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img51.png"
|
|
|
|
|
ALT="$\Omega^k_i$"></SPAN>.
|
|
|
|
|
The classical AS preconditioner <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$M^k_{AS}$"></SPAN> is defined as
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER" CLASS="mathdisplay">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="218" HEIGHT="59" BORDER="0"
|
|
|
|
|
SRC="img59.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
( M^k_{AS} )^{-1} = \sum_{i=1}^{m_k} P_i^k (A_i^k)^{-1} R_i^{k},
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
|
where <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img60.png"
|
|
|
|
|
ALT="$A_i^k$"></SPAN> is supposed to be nonsingular. We observe that an approximate
|
|
|
|
|
inverse of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img60.png"
|
|
|
|
|
ALT="$A_i^k$"></SPAN> is usually considered instead of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img61.png"
|
|
|
|
|
ALT="$(A_i^k)^{-1}$"></SPAN>.
|
|
|
|
|
The setup of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$M^k_{AS}$"></SPAN> during the multilevel build phase
|
|
|
|
|
involves
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<UL>
|
|
|
|
|
<LI>the definition of the index subspaces <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img62.png"
|
|
|
|
|
ALT="$\Omega_i^k$"></SPAN> and of the corresponding
|
|
|
|
|
operators <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img63.png"
|
|
|
|
|
ALT="$R_i^k$"></SPAN> (and <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img64.png"
|
|
|
|
|
ALT="$P_i^k$"></SPAN>);
|
|
|
|
|
</LI>
|
|
|
|
|
<LI>the computation of the submatrices <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img60.png"
|
|
|
|
|
ALT="$A_i^k$"></SPAN>;
|
|
|
|
|
</LI>
|
|
|
|
|
<LI>the computation of their inverses (usually approximated
|
|
|
|
|
through some form of incomplete factorization).
|
|
|
|
|
</LI>
|
|
|
|
|
</UL><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
|
The computation of <!-- MATH
|
|
|
|
|
$z^k=M^k_{AS}w^k$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img65.png"
|
|
|
|
|
ALT="$z^k=M^k_{AS}w^k$"></SPAN>, with <!-- MATH
|
|
|
|
|
$w^k \in \mathbb{R}^{n_k}$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img66.png"
|
|
|
|
|
ALT="$w^k \in \mathbb{R}^{n_k}$"></SPAN>, during the
|
|
|
|
|
multilevel application phase, requires
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<UL>
|
|
|
|
|
<LI>the restriction of <SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img67.png"
|
|
|
|
|
ALT="$w^k$"></SPAN> to the subspaces <!-- MATH
|
|
|
|
|
$\mathbb{R}^{n_{k,i}}$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img68.png"
|
|
|
|
|
ALT="$\mathbb{R}^{n_{k,i}}$"></SPAN>,
|
|
|
|
|
i.e. <!-- MATH
|
|
|
|
|
$w_i^k = R_i^{k} w^k$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img69.png"
|
|
|
|
|
ALT="$w_i^k = R_i^{k} w^k$"></SPAN>;
|
|
|
|
|
</LI>
|
|
|
|
|
<LI>the computation of the vectors <!-- MATH
|
|
|
|
|
$z_i^k=(A_i^k)^{-1} w_i^k$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img70.png"
|
|
|
|
|
ALT="$z_i^k=(A_i^k)^{-1} w_i^k$"></SPAN>;
|
|
|
|
|
</LI>
|
|
|
|
|
<LI>the prolongation and the sum of the previous vectors,
|
|
|
|
|
i.e. <!-- MATH
|
|
|
|
|
$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$
|
|
|
|
|
-->
|
|
|
|
|
<SPAN CLASS="MATH"><IMG
|
|
|
|
|
WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img71.png"
|
|
|
|
|
ALT="$z^k = \sum_{i=1}^{m_k} P_i^k z_i^k$"></SPAN>.
|
|
|
|
|
</LI>
|
|
|
|
|
</UL><BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">
|
|
|
|
|
Variants of the classical AS method, which use modifications of the
|
|
|
|
|
restriction and prolongation operators, are also implemented in MLD2P4.
|
|
|
|
|
Among them, the Restricted AS (RAS) preconditioner usually
|
|
|
|
|
outperforms the classical AS preconditioner in terms of convergence
|
|
|
|
|
rate and of computation and communication time on parallel distributed-memory
|
|
|
|
|
computers, and is therefore the most widely used among the AS
|
|
|
|
|
preconditioners [<A
|
|
|
|
|
HREF="node36.html#CAI_SARKIS">6</A>].
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<P>
|
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE">Direct solvers based on sparse LU factorizations, implemented in the
|
|
|
|
|
third-party libraries reported in Section <A HREF="node8.html#sec:third-party">3.2</A>, can be applied
|
|
|
|
|
as coarsest-level solvers by MLD2P4. Native inexact solvers based on
|
|
|
|
|
incomplete LU factorizations, as well as Jacobi, hybrid (forward) Gauss-Seidel,
|
|
|
|
|
and block Jacobi preconditioners are also available. Direct solvers usually
|
|
|
|
|
lead to more effective preconditioners in terms of algorithmic scalability;
|
|
|
|
|
however, this does not guarantee parallel efficiency.
|
|
|
|
|
</BIG></BIG></BIG>
|
|
|
|
|
<P>
|
|
|
|
|
<BIG CLASS="LARGE"><BIG CLASS="LARGE"><BIG CLASS="LARGE"></BIG></BIG></BIG>
|
|
|
|
|
<DIV CLASS="navigation"><HR>
|
|
|
|
|
<!--Navigation Panel-->
|
|
|
|
|
<A NAME="tex2html271"
|
|
|
|
|
HREF="node16.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A>
|
|
|
|
|
<A NAME="tex2html267"
|
|
|
|
|
HREF="node12.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A>
|
|
|
|
|
<A NAME="tex2html263"
|
|
|
|
|
HREF="node14.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A>
|
|
|
|
|
<A NAME="tex2html269"
|
|
|
|
|
HREF="node2.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A>
|
|
|
|
|
<BR>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html272"
|
|
|
|
|
HREF="node16.html">Getting Started</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html268"
|
|
|
|
|
HREF="node12.html">Multigrid Background</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html264"
|
|
|
|
|
HREF="node14.html">Smoothed Aggregation</A>
|
|
|
|
|
<B> <A NAME="tex2html270"
|
|
|
|
|
HREF="node2.html">Contents</A></B> </DIV>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
