update ex15.c

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index c66f8c71ae1ec2704b0db181df987640f4a3d97f..d8fc84bef75fe57cc77ac89d7b81a3fe49d30780 100644 (file)
--- 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 $x^0$
   \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 $R=AS$
     \State $S_{k~mod~s}=x^k$ 
     \If {$k$ mod $s=0$ {\bf and} not convergence}
       \State Compute dense matrix $R=AS$


@@ -690,12 +690,28 @@ torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
 \end{center}
 \end{table}
 
 \end{center}
 \end{table}
 

-In  table~\ref{tab:02}, some  experiments comparing  the sovling  of  the linear
+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,
 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.
+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}
 
 
 \begin{table}


@@ -703,7 +719,8 @@ solver and the same preconditionner is used.
 \begin{tabular}{|c|c|r|r|r|r|} 
 \hline
 
 \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  \\
        &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
 
 crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\


@@ -711,22 +728,43 @@ 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 \\
 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}
 \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}
 
 
 \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. cores & precond   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} &  \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ 
+\cline{3-8}
+             &                       & 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 \\
+  16,384     & sor                   & 2,852.14 & 216,240   & 418.46 & 21,690  & 505.26 & 23,970 & 6.82 \\
+\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*}
+

 
 
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