X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/9c834284514e7988ab1e0780d1b58a9a9f317751..16f0ca498711e0af6a2de8200c2fbee0e5f8810e:/paper.tex

diff --git a/paper.tex b/paper.tex
index 087ff6e..35d9ed7 100644
--- a/paper.tex
+++ b/paper.tex
@@ -353,6 +353,7 @@
 \usepackage{algpseudocode}
 \usepackage{amsmath}
 \usepackage{amssymb}
+\usepackage{multirow}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
@@ -636,8 +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 $R=AS$
@@ -663,6 +664,81 @@ reused with the new values of the residuals.
 \section{Experiments using petsc}
 \label{sec:04}
 
+
+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. For all  the matrices, the name, the field,  the number of rows
+and the number of nonzero elements is given.
+
+\begin{table}
+\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 \\
+epb3               & Thermal problem   & 84,617  & 463,625  \\
+atmosmodj          & Computational fluid dynamics  & 1,270,432 & 8,814,880 \\
+bfwa398            & Electromagnetics problem & 398 & 3,678 \\
+torso3             & 2D/3D problem & 259,156 & 4,429,042 \\
+\hline
+
+\end{tabular}
+\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
+\label{tab:01}
+\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}
+\begin{center}
+\begin{tabular}{|c|c|r|r|r|r|} 
+\hline
+
+ \multirow{2}{*}{Matrix name}  & Solver /   & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage} \\ 
+\cline{3-6}
+       &  precond             & Time  & \# Iter.  & Time  & \# Iter.  \\\hline \hline
+
+crashbasis         & gmres / none             &  15.65     & 518  &  14.12 & 450  \\
+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  & 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.}
+\label{tab:02}
+\end{center}
+\end{table}
+
+
+
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