X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/Krylov_multi.git/blobdiff_plain/d9105f3cc518086be070a250845823a6ec0f7b4a..774e3ae76fae2fcab8efe8bf6bc2b61c0da65e09:/krylov_multi.tex?ds=sidebyside

diff --git a/krylov_multi.tex b/krylov_multi.tex
index 0ff9175..8685bd2 100644
--- a/krylov_multi.tex
+++ b/krylov_multi.tex
@@ -206,7 +206,7 @@ is solved  independently by a cluster of  processors and communication
 are required  to update the  right-hand side vectors $Y_l$,  such that
 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     for    solving     the
+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
 parallel on a cluster of processors.
@@ -234,7 +234,10 @@ S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 \label{sec03:eq04}
 \end{equation}
 where   for  $j\in\{1,\ldots,s\}$,  $x^j=[X_1^j,\ldots,X_L^j]$   is  a
-solution of the  global linear system. The advantage  of such a  Krylov subspace is  that we need  neither an orthogonal basis  nor synchronizations between  the different clusters to generate this basis.
+solution of the  global linear system. The advantage  of such a Krylov
+subspace   is  that   we  need   neither  an   orthogonal   basis  nor
+synchronizations  between  the  different  clusters to  generate  this
+basis.
 
 The  multisplitting   method  is  periodically   restarted  every  $s$
 iterations  with   a  new  initial   guess  $\tilde{x}=S\alpha$  which
@@ -256,12 +259,12 @@ which is associated with the least squares problem
 \text{minimize}~\|b-R\alpha\|_2,
 \label{sec03:eq07}
 \end{equation}  
-where  $R^T$  denotes the  transpose  of  the  matrix $R$.  Since  $R$
-(i.e.  $AS$) and  $b$  are  split among  $L$  clusters, the  symmetric
-positive    definite    system~(\ref{sec03:eq06})    is   solved    in
-parallel. Thus, an  iterative method would be more  appropriate than a
-direct  one for  solving this  system. We  use the  parallel conjugate
-gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
+where $R^T$ denotes the transpose  of the matrix $R$.  Since $R$ (i.e.
+$AS$) and  $b$ are  split among $L$  clusters, the  symmetric positive
+definite  system~(\ref{sec03:eq06}) is  solved in  parallel.  Thus, an
+iterative method would be more  appropriate than a direct one to solve
+this system.  We use  the parallel conjugate  gradient method  for the
+normal equations CGNR~\cite{S96,refCGNR}.
 
 \begin{algorithm}[!t]
 \caption{A two-stage linear solver with inner iteration GMRES method}
@@ -301,16 +304,16 @@ gradient method for the normal equations CGNR~\cite{S96,refCGNR}.
 \label{algo:01}
 \end{algorithm}
 
-The main  key points  of the multisplitting  method for  solving large
-sparse  linear  systems are  given  in Algorithm~\ref{algo:01}.   This
+The  main key points  of the  multisplitting method  to solve  a large
+sparse  linear  system  are  given in  Algorithm~\ref{algo:01}.   This
 algorithm is based on a two-stage method with a minimization using the
 GMRES iterative method as an  inner solver. It is executed in parallel
 by  each cluster  of processors.   The matrices  and vectors  with the
 subscript  $l$ represent  the local  data for  the cluster  $l$, where
 $l\in\{1,\ldots,L\}$. The two-stage solver uses two different parallel
-iterative algorithms: the GMRES method for solving each splitting on a
+iterative algorithms:  the GMRES method  to solve each splitting  on a
 cluster of processors, and the CGNR method executed in parallel by all
-clusters for  minimizing the function  error over the  Krylov subspace
+clusters  to minimize  the  function error  over  the Krylov  subspace
 spanned by  $S$.  The  algorithm requires two  global synchronizations
 between the $L$  clusters. The first one is  performed at line~$12$ in
 Algorithm~\ref{algo:01}  to exchange  the local  values of  the vector