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 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
%Doing many experiments with many cores is not easy and requires to access to a supercomputer with several hours for developing a code and then improving it.
In the following we present some experiments we could achieved out on the Hector
-architecture, the previous UK's high-end computing resource, funded by the UK
-Research Councils. This is a Cray XE6 supercomputer, equipped with two 16-core
-AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected with a 3D
-torus.
+architecture, a UK's high-end computing resource, funded by the UK Research
+Councils~\cite{hector}. This is a Cray XE6 supercomputer, equipped with two
+16-core AMD Opteron 2.3 Ghz and 32 GB of memory. Machines are interconnected
+with a 3D torus.
Table~\ref{tab1} shows the result of the experiments. The first column shows
the size of the 3D Poisson problem. The size is chosen in order to have
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\\
