In order to accelerate the convergence, the outer iteration periodically applies
a least-square minimization on the residuals computed by the inner solver. The
-inner solver is a Krylov based solver which does not required to be changed.
+inner solver is based on a Krylov method which does not require to be changed.
At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
iterations, using an iterative method restarting with the previous solution. For
the GMRES every 30 iterations, $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
chosen to minimize the least-squares problem with the following parameters:
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
-$1e-10$ (i.e. ). Those experiments
-have been performed on a Intel(R) Core(TM) i7-3630QM CPU @ 2.40GHz with the
-version 3.5.1 of PETSc.
+$\epsilon_{tsarm}=1e-10$. 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.
+different preconditioner is used. With TSARM, the same solver and the same
+preconditionner is used. This Table shows that TSARM 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}
-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:
+In order to perform larger experiments, we have tested some example application
+of PETSc. 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 an operator using a finite
difference scheme. The diagonal 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
+ used in many physical phenomena, for example, heat and fluid flow, wave
propagation...
\item ex54 is another example based on 2D problem discretized with quadrilateral
finite elements. For this example, the user can define the scaling of material
chosen because they are scalable with many cores. We have tested other problem
but they are not scalable with many cores.
+In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
+{\bf Description of preconditioners}
\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & precond & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSARM CGLS} & \multicolumn{2}{c|}{TSARM LSQR} & best gain \\
+ nb. cores & precond & \multicolumn{2}{c|}{FGMRES} & \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 \\
\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.}
+\caption{Comparison of FGMRES and TSARM with FGMRES for example ex15 of PETSc with two preconditioner (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
\label{tab:03}
\end{center}
\end{table*}
+Table~\ref{tab:03} shows the execution times and the number of iterations of
+example ex15 of PETSc on the Juqueen architecture. Differents number of cores
+are studied rangin from 2,048 upto 16,383. Two preconditioners have been
+tested. For those experiments, the number of components (or unknown of the
+problems) per processor is fixed to 25,000. This number can seem relatively
+small. In fact, for some applications that need a lot of memory, the number of
+components per processor requires sometimes to be small.
+
+In this Table, we can notice that TSARM is always faster than FGMRES. The last
+column shows the ratio between FGMRES and the best version of TSARM according to
+the minimization procedure: CGLS or LSQR.
+
\begin{figure}
\centering
