Relecture

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -601,22 +601,22 @@ is summarized while intended perspectives are provided.
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 
 %%%*********************************************************
 \section{Related works}
 \label{sec:02} 

-Krylov subspace iteration methods have increasingly become useful and successful
-techniques  for  solving  linear,  nonlinear systems  and  eigenvalue  problems,
-especially      since       the      increase      development       of      the
+Krylov subspace iteration methods have increasingly become key
+techniques  for  solving  linear and nonlinear systems,  or  eigenvalue  problems,
+especially      since       the      increasing      development       of      

 preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of
 preconditioners~\cite{Saad2003,Meijerink77}.  One reason  of  the popularity  of

-these methods is their generality, simplicity and efficiency to solve systems of
+these methods is their generality, simplicity, and efficiency to solve systems of

 equations arising from very large and complex problems.
 
 GMRES is one of the most  widely used Krylov iterative method for solving sparse
 equations arising from very large and complex problems.
 
 GMRES is one of the most  widely used Krylov iterative method for solving sparse

-and large  linear systems. It  is developed by  Saad and al.~\cite{Saad86}  as a
+and large  linear systems. It  has been developed by  Saad \emph{et al.}~\cite{Saad86}  as a

 generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and
 generalized  method to  deal with  unsymmetric and  non-Hermitian  problems, and

-indefinite symmetric problems too. In its original version called full GMRES, it
+indefinite symmetric problems too. In its original version called full GMRES, this algorithm

 minimizes the residual over the  current Krylov subspace until convergence in at
 minimizes the residual over the  current Krylov subspace until convergence in at

-most $n$ iterations,  where $n$ is the  size of the sparse matrix.  It should be
-noticed that full GMRES is too expensive in the case of large matrices since the
+most $n$ iterations,  where $n$ is the  size of the sparse matrix.  
+Full GMRES is however too much expensive in the case of large matrices, since the

 required orthogonalization  process per  iteration grows quadratically  with the
 required orthogonalization  process per  iteration grows quadratically  with the

-number of iterations. For that reason, in practice GMRES is restarted after each
+number of iterations. For that reason, GMRES is restarted in practice after each

 $m\ll n$ iterations to avoid the  storage of a large orthonormal basis. However,
 the  convergence behavior  of the  restarted GMRES,  called GMRES($m$),  in many
 cases depends quite critically on  the value of $m$~\cite{Huang89}. Therefore in
 $m\ll n$ iterations to avoid the  storage of a large orthonormal basis. However,
 the  convergence behavior  of the  restarted GMRES,  called GMRES($m$),  in many
 cases depends quite critically on  the value of $m$~\cite{Huang89}. Therefore in


@@ -1138,12 +1138,12 @@ Concerning the  experiments some  other remarks are  interesting.
   examples,  we also obtained  similar gain  between GMRES  and TSIRM  but those
   examples are  not scalable with many  cores. In general, we  had some problems
   with more than $4,096$ cores.
   examples,  we also obtained  similar gain  between GMRES  and TSIRM  but those
   examples are  not scalable with many  cores. In general, we  had some problems
   with more than $4,096$ cores.

-\item We have tested many iterative  solvers available in PETSc.  In fast, it is
+\item We have tested many iterative  solvers available in PETSc.  In fact, it is

   possible to use most of them with TSIRM. From our point of view, the condition
   to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart
   possible to use most of them with TSIRM. From our point of view, the condition
   to  use  a  solver inside  TSIRM  is  that  the  solver  must have  a  restart

-  feature. More  precisely, the solver must  support to be  stoped and restarted
+  feature. More  precisely, the solver must  support to be  stopped and restarted

   without decrease its  converge. That is why  with GMRES we stop it  when it is
   without decrease its  converge. That is why  with GMRES we stop it  when it is

-  naturraly  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate
+  naturally  restarted (i.e.  with  $m$ the  restart parameter).   The Conjugate

   Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
   so they  are not  efficient.  They  will converge with  TSIRM but  not quickly
   because if  we compare  a normal CG  with a CG  for which  we stop it  each 16
   Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
   so they  are not  efficient.  They  will converge with  TSIRM but  not quickly
   because if  we compare  a normal CG  with a CG  for which  we stop it  each 16