Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/Krylov_multi

[Krylov_multi.git] / krylov_multi.tex

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 \maketitle
 
 
 \maketitle
 
 

+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+

 \begin{abstract}
 \begin{abstract}

-In this paper  we revist the krylov multisplitting  algorithm presented in [ref]
-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
+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}
 
 classical GMRES both in terms of number of iterations and execution times.
 \end{abstract}
 

+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+

 \section{Introduction}
 
 \section{Introduction}
 

-Iterative methods used  to solve large sparse linear systems  of the form $Ax=b$
-because they are easier to parallelize than direct ones.
+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.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{A two-stage method with a minimization}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+\bibliographystyle{plain}
+\bibliography{biblio}

 
 \end{document}
 
 \end{document}