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[Krylov_multi.git] / krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 98b8949c4b23eec6282fc86269dd6206df71ac0d..2d0bc6994339e973f66571ac17f0008f0b7b822a 100644 (file)
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -75,11 +75,14 @@ traditional  multisplitting  method  that  suffer  from  slow  convergence,  as
 proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
 drastically improve the convergence.
 
 proposed  in~\cite{huang1993krylov},  the  use  of a  minimization  process  can
 drastically improve the convergence.
 

+The paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. The, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem.
+

 
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-\section{Related works and presention of the multisplitting method}
+\section{Related works and presentation of the multisplitting method}
+\label{sec:02}

 A general framework  for studying parallel multisplitting has  been presented in~\cite{o1985multi}
 by O'Leary and White. Convergence conditions are given for the
 most general case.  Many authors improved multisplitting algorithms by proposing
 A general framework  for studying parallel multisplitting has  been presented in~\cite{o1985multi}
 by O'Leary and White. Convergence conditions are given for the
 most general case.  Many authors improved multisplitting algorithms by proposing


@@ -142,6 +145,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
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 \section{A two-stage method with a minimization}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{A two-stage method with a minimization}

+\label{sec:03}

 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. The multisplitting of this linear system is defined as follows
 \begin{equation}
 \left\{
 Let $Ax=b$ be a given large and sparse linear system of $n$ equations to solve in parallel on $L$ clusters of processors, physically adjacent or geographically distant, where $A\in\mathbb{R}^{n\times n}$ is a square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. The multisplitting of this linear system is defined as follows
 \begin{equation}
 \left\{


@@ -253,6 +257,7 @@ The main key points of our Krylov multisplitting method to solve a large sparse
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experiments}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experiments}

+\label{sec:04}

 In order to illustrate  the interest  of our algorithm. We have  compared our
 algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
 situations.  We have chosen to focus on only one problem which is very simple to
 In order to illustrate  the interest  of our algorithm. We have  compared our
 algorithm  with  the  GMRES  method  which  is a very  well  used  method  in  many
 situations.  We have chosen to focus on only one problem which is very simple to