solvers. However, most of the good preconditioners are not scalable when
thousands of cores are used.
-Traditional iterative solvers have global synchronizations that penalize the
-scalability. Two possible solutions consists either in using asynchronous
-iterative methods~\cite{ref18} or to use multisplitting algorithms. In this
-paper, we will reconsider the use of a multisplitting method. In opposition to
-traditional multisplitting method that suffer from slow convergence, as
-proposed in~\cite{huang1993krylov}, the use of a minimization process can
-drastically improve the convergence.
+%Traditional iterative solvers have global synchronizations that penalize the
+%scalability. Two possible solutions consists either in using asynchronous
+%iterative methods~\cite{ref18} or to use multisplitting algorithms. In this
+%paper, we will reconsider the use of a multisplitting method. In opposition to
+%traditional multisplitting method that suffer from slow convergence, as
+%proposed in~\cite{huang1993krylov}, the use of a minimization process can
+%drastically improve the convergence.
+
+Traditional parallel iterative solvers are based on fine-grain computations that frequently require data exchanges between computing nodes and have global synchronizations that penalize the scalability. Particularly, they are more penalized on large scale architectures or on distributed platforms composed of distant clusters interconnected by a high-latency network. It is therefore imperative to develop coarse-grain based algorithms to reduce the communications in the parallel iterative solvers. Two possible solutions consists either in using asynchronous iterative methods~\cite{ref18} or to use multisplitting algorithms. In this paper, we will reconsider the use of a multisplitting method. In opposition to 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 present paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. Then, 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 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
%%%%%%%%%%%%%%%%%%%%%%%%
\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\{
%%%%%%%%%%%%%%%%%%%%%%%%
\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
preconditioning techniques for Krylov iterative methods and multisplitting
methods with overlapping blocks.
+\section{Acknowledgement}
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
%Other applications (=> other matrices)\\
%Larger experiments\\
