X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/94bff40349bba73bd034821805ca870d73d148c9..fd78ced113c137f5f83f7aea69a10961467aea3e:/paper.tex

diff --git a/paper.tex b/paper.tex
index caf88ea..16b2de5 100644
--- a/paper.tex
+++ b/paper.tex
@@ -885,21 +885,21 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \end{center}
 \end{table}
 Chosen parameters  are detailed below.   
-We have  chosen  to stop  the  GMRES every  30
+We have  stopped  the  GMRES every  30
 iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default 
-setting of GMRES.  $s$, for its part, has been set to 8. CGLS is chosen
-to   minimize  the   least-squares  problem   with  the   following  parameters:
+setting of GMRES.  $s$, for its part, has been set to 8. CGLS 
+ minimizes  the   least-squares  problem   with  parameters
 $\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$.  The external precision is set to
-$\epsilon_{tsirm}=1e-10$.  Those  experiments have been performed  on a Intel(R)
-Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
+$\epsilon_{tsirm}=1e-10$.  These  experiments have been performed  on an Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version  of PETSc.
 
 
-In  Table~\ref{tab:02}, some  experiments comparing  the solving  of  the linear
-systems obtained with the previous matrices  with a GMRES variant and with TSIRM
-are given. In the  second column, it can be noticed that  either GMRES or FGMRES
-(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system.  According
-to  the matrices,  different preconditioners  are  used.  With  TSIRM, the  same
-solver and the  same preconditionner are used.  This Table  shows that TSIRM can
+Experiments comparing 
+a GMRES variant with TSIRM in the resolution of linear systems are given in  Table~\ref{tab:02}. 
+The  second column describes whether GMRES or FGMRES
+(Flexible GMRES~\cite{Saad:1993}) has been used for linear systems solving.  
+Different preconditioners  have been used according to the matrices.  With  TSIRM, the  same
+solver and the  same preconditionner are used.  This table  shows that TSIRM can
 drastically reduce  the number of iterations  to reach the  convergence when the
 number of iterations for  the normal GMRES is more or less  greater than 500. In
 fact this also depends on two parameters: the number of iterations to stop GMRES
@@ -924,7 +924,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
-\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
 \label{tab:02}
 \end{center}
 \end{table}
@@ -934,10 +934,10 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 
 In order to perform larger experiments, we have tested some example applications
-of  PETSc. Those  applications are  available in  the \emph{ksp}  part  which is
+of  PETSc. Those  applications are  available in  the \emph{ksp}  part,  which is
 suited for scalable linear equations solvers:
 \begin{itemize}
-\item ex15  is an example  which solves in  parallel an operator using  a finite
+\item ex15  is an example  that solves in  parallel an operator using  a finite
   difference  scheme.   The  diagonal  is  equal to  4  and  4  extra-diagonals
   representing the neighbors in each directions  are equal to -1. This example is
   used  in many  physical phenomena, for  example, heat  and fluid  flow, wave