X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/774e3ae76fae2fcab8efe8bf6bc2b61c0da65e09..09702354d347f9bf651fba24d04f262c757e2cc5:/krylov_multi.tex

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 8685bd2..7a7e809 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -14,6 +14,7 @@
 
 
 \title{A scalable multisplitting algorithm for solving large sparse linear systems} 
+\date{}
 
 
 
@@ -28,7 +29,7 @@
 
 
 \begin{abstract}
-In  this  paper we  revist  the  krylov  multisplitting algorithm  presented  in
+In  this paper  we  revisit  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
@@ -208,7 +209,7 @@ the  vectors  $X_i$  represent   the  data  dependencies  between  the
 clusters. In this work,  we use the parallel GMRES method~\cite{ref34}
 as     an     inner      iteration     method     to     solve     the
 sub-systems~(\ref{sec03:eq03}).  It  is a well-known  iterative method
-which  gives good performances  for solving  sparse linear  systems in
+which  gives good performances  to solve  sparse linear  systems in
 parallel on a cluster of processors.
 
 It should be noted that  the convergence of the inner iterative solver
@@ -327,6 +328,36 @@ synchronizations   by   using   the   MPI   collective   communication
 subroutines.
 
 
+\section{Experiments}
+
+In order  to illustrate  the interest  of our algorithm.   We have  compared our
+algorithm  with  the  GMRES  method  which  a very  well  used  method  in  many
+situations.  We have chosen to focus on only one problem which is very simple to
+implement: a 3 dimension Poisson problem.
+
+\begin{equation}
+\left\{
+                \begin{array}{ll}
+                  \nabla u&=f \mbox{~in~} \omega\\
+                  u &=0 \mbox{~on~}  \Gamma=\partial \omega
+                \end{array}
+              \right.
+\end{equation}
+
+After discretization, with a finite  difference scheme, a seven point stencil is
+used. It  is well-known that the  spectral radius of  matrices representing such
+problems are very close to 1.  Moreover, the larger the number of discretization
+points is,  the closer to 1  the spectral radius  is.  Hence, to solve  a matrix
+obtained for  a 3D Poisson  problem, the number  of iterations is high.  Using a
+preconditioner  it  is   possible  to  reduce  the  number   of  iterations  but
+preconditioners are not scalable when using many cores.
+
+\section{Conclusion and perspectives}
+
+Other applications (=> other matrices)\\
+Larger experiments\\
+Async\\
+Overlapping
 
 
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