X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/7ff6342cfd0d2fa0e5df79322a88117ece68880e..b0a581c5855a3dda28df994ccde7482913b7be1e:/paper.tex

diff --git a/paper.tex b/paper.tex
index 36298aa..374f147 100644
--- a/paper.tex
+++ b/paper.tex
@@ -637,8 +637,8 @@ direct method in the parallel context.
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
   \State Set the initial guess $x^0$
-  \For {$k=1,2,3,\ldots$ until convergence}
-    \State Solve iteratively $Ax^k=b$
+  \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv}
+    \State Solve iteratively $Ax^k=b$  \label{algo:solve}
     \State $S_{k~mod~s}=x^k$ 
     \If {$k$ mod $s=0$ {\bf and} not convergence}
       \State Compute dense matrix $R=AS$
@@ -668,7 +668,8 @@ reused with the new values of the residuals.
 In order to see the influence of our algorithm with only one processor, we first
 show  a comparison  with the  standard version  of GMRES  and our  algorithm. In
 table~\ref{tab:01},  we  show  the  matrices  we  have used  and  some  of  them
-characteristics.
+characteristics. For all  the matrices, the name, the field,  the number of rows
+and the number of nonzero elements is given.
 
 \begin{table}
 \begin{center}
@@ -689,7 +690,28 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \end{center}
 \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     GMRES    every     30    iterations     (line     \ref{algo:solve}    in
+Algorithm~\ref{algo:01}).   $s$ is  set to  8. CGLS  is chosen  to  minimize the
+least-squares  problem.  Two  conditions  are  used to  stop  CGLS,  either  the
+precision is under $1e-40$ or the  number of iterations is greater to $20$.  The
+external   precision    is   set    to   $1e-10$   (line    \ref{algo:conv}   in
+Algorithm~\ref{algo:01}).  Those  experiments have been performed  on a Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 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 out 2
+stage algorithm are  given. In the second column, it can  be noticed that either
+gmres or fgmres is used to  solve the linear system.  According to the matrices,
+different preconditioner is  used.  With the 2 stage  algorithm, the same solver
+and  the same  preconditionner  is used.   This  Table shows  that  the 2  stage
+algorithm  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 tow parameters: the  number of
+iterations  to  stop  GMRES  and   the  number  of  iterations  to  perform  the
+minimization.
 
 
 \begin{table}
@@ -697,30 +719,52 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
- \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage} \\
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ 
+\cline{3-6}
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
 crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
-parabolic\_fem     & gmres / ilu           &  2152  & ?? & 724 & ?? \\
+parabolic\_fem     & gmres / ilu           & 1009.94   & 7573 & 401.52 & 2970 \\
 epb3               & fgmres / sor             &  8.67     & 600  &  8.21 & 540  \\
 atmosmodj          &  fgmres / sor & 104.23  & 451 & 88.97 & 366  \\
 bfwa398            & gmres / none  & 1.42 & 9612 & 0.28 & 1650 \\
-torso3             & fgmres/sor  & 565  & 37.70 & 34.97 & 510 \\
+torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
-\caption{Comparison of GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
 \label{tab:02}
 \end{center}
 \end{table}
 
 
-Param : retart 30 iters
-cols = 8
-iter cgls = 20
-cgls prec = 1e-40
-prec = 1e-10
-Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz
+
+
+Larger experiments ....
+
+\begin{table*}
+\begin{center}
+\begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
+\hline
+
+   & nb. comp.   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+\cline{3-8}
+  nb. cores&    precond             & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. & \\\hline \hline
+
+  4,096      & mg                    & 562.25   & 25,170    & 97.23  & 3,990   & 89.71  & 3,630  & 6.27 \\
+  4,096      & sor                   & 912.12   & 70,194    & 145.57 & 9,750   & 168.97 & 10,980 & 6.26 \\
+  8,192      & mg                    & 917.02   & 40,290    & 148.81 & 5,730   & 143.03 & 5,280  & 6.41 \\
+  8,192      & sor                   & 1,404.53 & 106,530   & 212.55 & 12,990  & 180.97 & 10,470 & 7.76 \\
+  16,384     & mg                    & 1,430.56 & 63,930    & 237.17 & 8,310   & 244.26 & 7,950  & 6.03 \\
+
+\hline
+
+\end{tabular}
+\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex15 of Petsc with 25000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  time is expressed in seconds.}
+\label{tab:03}
+\end{center}
+\end{table*}
+
 
 
 %%%*********************************************************