X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/aeb5859d1996432c60c4346b9f961298caa11f5a..124fbe31852f69cefa44e364e767c5e0e07ef670:/paper.tex

diff --git a/paper.tex b/paper.tex
index b1c6f59..fb68702 100644
--- a/paper.tex
+++ b/paper.tex
@@ -431,9 +431,9 @@ convergence of Krylov iterative methods,  typically those of GMRES variants. The
 principle of  our approach  is to  build an external  iteration over  the Krylov
 method  and to  save  the current  residual  frequently (for  example, for  each
 restart of GMRES). Then after a given number of outer iterations, a minimization
-step is applied on the matrix composed of the save residuals in order to compute
-a  better solution and  make a  new iteration  if necessary.  We prove  that our
-method  has the  same  convergence property  than  the inner  method used.  Some
+step  is applied  on the  matrix composed  of the  saved residuals  in  order to
+compute a better solution and make  a new iteration if necessary.  We prove that
+our method has  the same convergence property than the  inner method used.  Some
 experiments using up  to 16,394 cores show that compared  to GMRES our algorithm
 can be around 7 times faster.
 \end{abstract}
@@ -793,7 +793,7 @@ minimization.
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
- \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ 
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} \\ 
 \cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
@@ -816,14 +816,25 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 
 Larger experiments ....\\
 
-Describe the problems ex15 and ex54
+In the following we describe the applications of PETSc we have experimented. Those applications are available in the ksp part which is suited for  scalable linear equations solvers:
+\begin{itemize}
+\item  ex15   is  an  example  which   solves  in  parallel   a  2D  homogeneous
+  Laplacian. Thediagonal is  equals to 4 and 4  extra-diagonals representing the
+  neighbors in  each directions  is equal to  -1. This  example is used  in many
+  physical phenomena , for exemple, heat and fluid flow, wave propagation...
+\item
+\end{itemize}
+
+
+
+
 
 \begin{table*}
 \begin{center}
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores & precond   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+  nb. cores & precond   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & mg                    & 403.49   & 18,210    & 73.89  & 3,060   & 77.84  & 3,270  & 5.46 \\
@@ -848,7 +859,7 @@ Describe the problems ex15 and ex54
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores & threshold   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain \\ 
 \cline{3-8}
              &                       & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
   2,048      & 8e-5                  & 108.88 & 16,560  & 23.06  &  3,630  & 22.79  & 3,630   & 4.77 \\
@@ -856,7 +867,7 @@ Describe the problems ex15 and ex54
   4,096      & 7e-5                  & 160.59 & 22,530  & 35.15  &  5,130  & 29.21  & 4,350   & 5.49 \\
   4,096      & 6e-5                  & 249.27 & 35,520  & 52.13  &  7,950  & 39.24  & 5,790   & 6.35 \\
   8,192      & 6e-5                  & 149.54 & 17,280  & 28.68  &  3,810  & 29.05  & 3,990  & 5.21 \\
-  8,192      & 5e-5                  & 792.11 & 109,590 & 76.83  &  10,470  & 65.20  & 9,030  & 12.14 \\
+  8,192      & 5e-5                  & 785.04 & 109,590 & 76.07  &  10,470  & 69.42 & 9,030  & 11.30 \\
   16,384     & 4e-5                  & 718.61 & 86,400 & 98.98  &  10,830  & 131.86  & 14,790  & 7.26 \\
 \hline
 
@@ -872,17 +883,17 @@ Describe the problems ex15 and ex54
 
 \begin{table*}
 \begin{center}
-\begin{tabular}{|r|r|r|r|r|r|r|r|} 
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
 \hline
 
-  nb. cores   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
-\cline{2-7}
-                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
-   512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070 &   \\
-   1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 &   \\
-   2048             & 919.62    & 31,470 & 237,52 & 8,040  & 194.22 & 6,510 &  \\
-   4096             & 405.60    & 28,380 & 111.67 & 7,590  & 91.72  & 6,510 &   \\
-   8192             & 785.04   & 109,590 & 76.07  & 10,470 & 69,42 & 9,030  & \\
+  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} &  \multicolumn{2}{c|}{TSARM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+\cline{2-7} \cline{9-11}
+                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
+   512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
+   1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
+   2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\
+   4096             & 405.60    & 28,380 & 111.67 & 7,590  & 91.72  & 6,510 & 4.42  & 1.22   &  .79     &   .84 \\
+   8192             & 785.04   & 109,590 & 76.07  & 10,470 & 69.42 & 9,030  & 11.30 &   .32  &   .58    &  .56 \\
 
 \hline