\section{Experiments using PETSc}\r
\label{sec:05}\r
\r
+%%NEW\r
+In this section four kinds of experiments have been performed. First, some experiments on real matrices issued from the sparse matrix florida have been achieved out. Second, some experiments in parallel with some linear problems are reported and analyzed. Third, some experiments in parallèle with som nonlinear problems are illustrated. Finally some parameters of TSIRM are studied in order to understand their influences.\r
+\r
+\r
+\subsection{Real matrices in sequential}\r
+%%ENDNEW\r
+\r
\r
In order to see the behavior of our approach when considering only one processor,\r
a first comparison with GMRES or FGMRES and the new algorithm detailed\r
\end{table*}\r
\r
\r
-\r
-\r
+%%NEW\r
+\subsection{Parallel linear problems}\r
+%%ENDNEW\r
\r
In order to perform larger experiments, we have tested some example applications\r
of PETSc. These applications are available in the \emph{ksp} part, which is\r
\r
\r
%%NEW\r
+\r
+\subsection{Nonlinear problems in parallel}\r
+\r
\begin{table*}[htbp]\r
\begin{center}\r
\begin{tabular}{|r|r|r|r|r|r|r|r|} \r
\end{table*}\r
\r
\r
+\subsection{Influcence of parameters for TSIRM}\r
+\r
%%ENDNEW\r
\r
%%%*********************************************************\r
%%%*********************************************************\r
%%%*********************************************************\r
\r
-A new two-stage iterative algorithm TSIRM has been proposed in this article,\r
-in order to accelerate the convergence of Krylov iterative methods.\r
-Our TSIRM proposal acts as a merger between Krylov based solvers and\r
-a least-squares minimization step.\r
-The convergence of the method has been proven in some situations, while \r
-experiments up to 16,394 cores have been led to verify that TSIRM runs\r
-5 or 7 times faster than GMRES.\r
+%%NEW\r
+In this paper a new two-stage algorithm TSIRM has been described. This method allows us to improve the convergence of Krylov iterative methods. It is based\r
+on a least-squares minimization step which uses the Krylov residuals.\r
+\r
\r
+We have implemented our code in PETSc in order to show that it is efficient and scalable. Some experiments with classical examples of PETSc for linear and nonlinear problems have been performed. We observed that TSIRM outperforms GMRES variants when the number of iterations is large. TSIRM is also scalable since we made some experiments with up to 16,394 cores.\r
\r
-For future work, the authors' intention is to investigate other kinds of\r
-matrices, problems, and inner solvers. In particular, the possibility \r
-to obtain a convergence of TSIRM in situations where the GMRES is divergent will be\r
-investigated. The influence of all parameters must be\r
-tested too, while other methods to minimize the residuals must be regarded. The\r
-number of outer iterations to minimize should become adaptive to improve the\r
-overall performances of the proposal. Finally, this solver will be implemented\r
-inside PETSc, which would be of interest as it would allows us to test\r
-all the non-linear examples and compare our algorithm with the other algorithm\r
-implemented in PETSc.\r
+We also observed that TSIRM is efficient with different preconditioners. The hypre preconditioner that is globally very efficient for many problems is also very time consuming. Consequently, sometimes using a less performent preconditioners may be a better solution. In that case, TSIRM is also more efficient than traditional Krylov methods.\r
\r
+{\bf A CHECKER !!}\r
+The influence of some important parameters of TSIRM have been studied. It can be noticed that they have a strong influence on the convergence speed\r
\r
-% conference papers do not normally have an appendix\r
+In future works, we plan to study other problems coming from different research areas. Other efficient Krylov optimisation methods as communication avoiding technique may be interesting to be investigated\r
+%%ENDNEW\r
\r
\r
\r
