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\title{A scalable multisplitting algorithm for solving large sparse linear systems} 



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\author{Raphaël Couturier \and Lilia Ziane Khodja}

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\begin{abstract}
In  this  paper we  revist  the  krylov  multisplitting algorithm  presented  in
\cite{huang1993krylov}  which  uses  a  scalar  method to  minimize  the  krylov
iterations computed by a multisplitting algorithm. Our new algorithm is simply a
parallel multisplitting algorithm  with few blocks of large  size and a parallel
krylov  minimization  is used  to  improve  the  convergence. Some  large  scale
experiments  with a 3D  Poisson problem  are presented.  They show  the obtained
improvements compared to a classical GMRES both in terms of number of iterations
and execution times.
\end{abstract}


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\section{Introduction}

Iterative methods are used to solve  large sparse linear systems of equations of
the form  $Ax=b$ because they are  easier to parallelize than  direct ones. Many
iterative  methods have  been proposed  and adpated  by many  researchers.  When
solving large  linear systems  with many cores,  iterative methods  often suffer
from  scalability  problems.    This  is  due  to  their   need  for  collective
communications to perform matrix-vector products and reduction operations.

%%%%% Lilia
% doit-on définir le principe et les préliminaires du multisplitting dans l'intro ou dans l'autre section? 
% valides-tu le titre de la 2eme section? celle que je voudrai rédiger.


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\section{A two-stage method with a minimization}


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