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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 based on
a  parallel multisplitting  algorithm  with few  blocks  of large  size using  a
parallel GMRES method inside each block and on a parallel krylov minimization in
order 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  adapted 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.
Preconditionners can be  used in order to increase  the convergence of iterative
solvers.   However, most  of the  good preconditionners  are not  sclalable when
thousands of cores are used.


A completer...
On ne peut pas parler de tout...

\section{Related works}


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
for  example  a  asynchronous  version  \cite{bru1995parallel}  and  convergence
condition  \cite{bai1999block,bahi2000asynchronous}   in  this  case   or  other
two-stage algorithms~\cite{frommer1992h,bru1995parallel}

In  \cite{huang1993krylov},  the  authors  proposed  a  parallel  multisplitting
algorithm in which all the tasks except  one are devoted to solve a sub-block of
the splitting  and to send their  local solution to  the first task which  is in
charge to  combine the vectors at  each iteration.  These vectors  form a Krylov
basis for  which the first tasks minimize  the error function over  the basis to
increase the convergence, then the other tasks receive the update solution until
convergence of the global system. 



In \cite{couturier2008gremlins}, the  authors proposed practical implementations
of multisplitting algorithms that take benefit from multisplitting algorithms to
solve large scale linear systems. Inner  solvers could be based on scalar direct
method with the LU method or scalar iterative one with GMRES.

%%%%% 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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