From b7730f90b7bda6c50247f984c438b3fb1fde97bf Mon Sep 17 00:00:00 2001
From: lilia <lilia@amazigh.bordeaux.inria.fr>
Date: Wed, 30 Apr 2014 17:34:31 +0200
Subject: [PATCH 1/1] v1
---
krylov_multi.tex | 5 +++++
1 file changed, 5 insertions(+)
diff --git a/krylov_multi.tex b/krylov_multi.tex
index 1dbd8cc..2d0bc69 100644
--- 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.
+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.
+
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\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
@@ -142,6 +145,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
%%%%%%%%%%%%%%%%%%%%%%%%
\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\{
@@ -253,6 +257,7 @@ The main key points of our Krylov multisplitting method to solve a large sparse
%%%%%%%%%%%%%%%%%%%%%%%%
\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
--
2.39.5