petite modif

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 15a45f01b040b1c1e94e37ab157aab8906124fd3..62927056fdd97597284db09396876b30ce329db5 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -428,16 +428,16 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-In  this article,  a  two-stage  iterative method is proposed to improve  the
-convergence of Krylov based iterative ones,  typically those of GMRES variants. The
+In  this article,  a  two-stage  iterative algorithm is proposed to improve  the
+convergence of Krylov based iterative methods,  typically those of GMRES variants. The

 principle of  the proposed approach  is to  build an external  iteration over  the Krylov
 method, and to  frequently store its current  residual   (at  each
 GMRES restart for instance). After a given number of outer iterations, a minimization
 step  is applied  on the  matrix composed by the  saved residuals,  in  order to
 principle of  the proposed approach  is to  build an external  iteration over  the Krylov
 method, and to  frequently store its current  residual   (at  each
 GMRES restart for instance). After a given number of outer iterations, a minimization
 step  is applied  on the  matrix composed by the  saved residuals,  in  order to

-compute a better solution while making  new iterations if required.  It is proven that
+compute a better solution and to make  new iterations if required.  It is proven that

 the proposal has  the same convergence properties than the  inner embedded method itself. 
 Experiments using up  to 16,394 cores also show that the proposed algorithm
 the proposal has  the same convergence properties than the  inner embedded method itself. 
 Experiments using up  to 16,394 cores also show that the proposed algorithm

-run around 7 times faster than GMRES.
+runs around 5 or 7 times faster than GMRES.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -759,16 +759,16 @@ Table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
 characteristics. For all  the matrices, the name, the field,  the number of rows
 and the number of nonzero elements is given.
 
 characteristics. For all  the matrices, the name, the field,  the number of rows
 and the number of nonzero elements is given.
 

-\begin{table*}[htbp]
+\begin{table}[htbp]

 \begin{center}
 \begin{tabular}{|c|c|r|r|r|} 
 \hline
 Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
 crashbasis         & Optimization      & 160,000  &  1,750,416  \\
 \begin{center}
 \begin{tabular}{|c|c|r|r|r|} 
 \hline
 Matrix name              & Field             &\# Rows   & \# Nonzeros   \\\hline \hline
 crashbasis         & Optimization      & 160,000  &  1,750,416  \\

-parabolic\_fem     & Computational fluid dynamics  & 525,825 & 2,100,225 \\
+parabolic\_fem     & Comput. fluid dynamics  & 525,825 & 2,100,225 \\

 epb3               & Thermal problem   & 84,617  & 463,625  \\
 epb3               & Thermal problem   & 84,617  & 463,625  \\

-atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
-bfwa398            & Electromagnetics problem & 398 & 3,678 \\
+atmosmodj          & Comput. fluid dynamics  & 1,270,432 & 8,814,880 \\
+bfwa398            & Electromagnetics pb & 398 & 3,678 \\

 torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \hline
 
 torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \hline
 


@@ -776,7 +776,7 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}
 \caption{Main characteristics of the sparse matrices chosen from the Davis collection}
 \label{tab:01}
 \end{center}

-\end{table*}
+\end{table}

 
 The following  parameters have been chosen  for our experiments.   As by default
 the restart  of GMRES is performed every  30 iterations, we have  chosen to stop
 
 The following  parameters have been chosen  for our experiments.   As by default
 the restart  of GMRES is performed every  30 iterations, we have  chosen to stop


@@ -882,23 +882,20 @@ number can seem relatively small. In fact, for some applications that need a lot
 of  memory, the  number of  components per  processor requires  sometimes  to be
 small.
 
 of  memory, the  number of  components per  processor requires  sometimes  to be
 small.
 

+
+

 In this Table, we  can notice that TSIRM is always faster  than FGMRES. The last
 column shows the ratio between FGMRES and the best version of TSIRM according to
 the minimization  procedure: CGLS or  LSQR. Even if  we have computed  the worst
 case  between CGLS  and LSQR,  it is  clear that  TSIRM is  alsways  faster than
 FGMRES. For this example, the  multigrid preconditionner is faster than SOR. The
 gain  between   TSIRM  and  FGMRES  is   more  or  less  similar   for  the  two
 In this Table, we  can notice that TSIRM is always faster  than FGMRES. The last
 column shows the ratio between FGMRES and the best version of TSIRM according to
 the minimization  procedure: CGLS or  LSQR. Even if  we have computed  the worst
 case  between CGLS  and LSQR,  it is  clear that  TSIRM is  alsways  faster than
 FGMRES. For this example, the  multigrid preconditionner is faster than SOR. The
 gain  between   TSIRM  and  FGMRES  is   more  or  less  similar   for  the  two

-preconditioners.
-
-In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
-Table~\ref{tab:01} is displayed.  It should  be noticed that for TSIRM, only the
-iterations of  the Krylov solver are  taken into account. Iterations  of CGLS or
-LSQR are  not recorded but they are  time-consuming. It can be  noticed that the
-number of iterations  per second of FMGRES is constant  whereas it decrease with
-TSIRM with both preconditioner. This can  be explained by the fact that when the
-number of core  increases the time for the minimization  step also increases but
-it  is  also  more efficient  to reduce the number of iterations.
-
+preconditioners.  Looking at the number  of iterations to reach the convergence,
+it is  obvious that TSIRM allows the  reduction of the number  of iterations. It
+should be noticed  that for TSIRM, in those experiments,  only the iterations of
+the Krylov solver  are taken into account.  Iterations of CGLS  or LSQR were not
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
+corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.

 
 \begin{figure}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -908,6 +905,17 @@ it  is  also  more efficient  to reduce the number of iterations.
 \end{figure}
 
 
 \end{figure}
 
 

+In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
+Table~\ref{tab:01}  is  displayed.   It  can  be  noticed  that  the  number  of
+iterations per second of FMGRES is  constant whereas it decrease with TSIRM with
+both preconditioner. This  can be explained by the fact that  when the number of
+core increases the time for the minimization step also increases but, generally,
+when  the number  of cores  increases,  the number  of iterations  to reach  the
+threshold also increases,  and, in that case, TSIRM is  more efficient to reduce
+the number of iterations. So, the overall benefit of using TSIRM is interesting.
+
+
+

 
 
 
 
 
 


@@ -935,7 +943,7 @@ it  is  also  more efficient  to reduce the number of iterations.
 \end{table*}
 
 
 \end{table*}
 
 

-
+In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.

 
 
 \begin{table*}[htbp]
 
 
 \begin{table*}[htbp]