\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 $R=AS$
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}
\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{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} \\
+\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}
\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
-
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