\r
\setcounter{page}{1}\r
\r
-\LRH{F. Wang et~al.}\r
+\LRH{R. Couturier, L. Ziane Khodja and C. Guyeux}\r
\r
-\RRH{Metadata Based Management and Sharing of Distributed Biomedical\r
-Data}\r
+\RRH{TSIRM: A Two-Stage Iteration with least-squares Residual Minimization algorithm}\r
\r
\VOL{x}\r
\r
\r
\BottomCatch\r
\r
-\PUBYEAR{2012}\r
+\PUBYEAR{2015}\r
\r
\subtitle{}\r
\r
\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}\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
\end{figure}\r
\r
\r
-Concerning the experiments some other remarks are interesting.\r
-\begin{itemize}\r
-\item We have tested other examples of PETSc/KSP (ex29, ex45, ex49). For all these\r
- examples, we have also obtained similar gains between GMRES and TSIRM but\r
- those examples are not scalable with many cores. In general, we had some\r
- problems with more than $4,096$ cores.\r
-\item We have tested many iterative solvers available in PETSc. In fact, it is\r
- possible to use most of them with TSIRM. From our point of view, the condition\r
- to use a solver inside TSIRM is that the solver must have a restart\r
- feature. More precisely, the solver must support to be stopped and restarted\r
- without decreasing its convergence. That is why with GMRES we stop it when it\r
- is naturally restarted (\emph{i.e.} with $m$ the restart parameter). The\r
- Conjugate Gradient (CG) and all its variants do not have ``restarted'' version\r
- in PETSc, so they are not efficient. They will converge with TSIRM but not\r
- quickly because if we compare a normal CG with a CG which is stopped and\r
- restarted every 16 iterations (for example), the normal CG will be far more\r
- efficient. Some restarted CG or CG variant versions exist and may be\r
- interesting to study in future works.\r
-\end{itemize}\r
-%%%*********************************************************\r
-%%%*********************************************************\r
-\r
-\r
-%%NEW\r
-\begin{table*}[htbp]\r
-\begin{center}\r
-\begin{tabular}{|r|r|r|r|r|r|r|r|} \r
-\hline\r
-\r
- nb. cores & \multicolumn{2}{c|}{FGMRES/ASM} & \multicolumn{2}{c|}{TSIRM CGLS/ASM} & gain& \multicolumn{2}{c|}{FGMRES/HYPRE} \\ \r
-\cline{2-5} \cline{7-8}\r
- & Time & \# Iter. & Time & \# Iter. & & Time & \# Iter. \\\hline \hline\r
- 512 & 5.54 & 685 & 2.5 & 570 & 2.21 & 128.9 & 9 \\\r
- 2048 & 14.95 & 1,560 & 4.32 & 746 & 3.48 & 335.7 & 9 \\\r
- 4096 & 25.13 & 2,369 & 5.61 & 859 & 4.48 & >1000 & -- \\\r
- 8192 & 44.35 & 3,197 & 7.6 & 1083 & 5.84 & >1000 & -- \\\r
-\r
-\hline\r
-\r
-\end{tabular}\r
-\caption{Comparison of FGMRES and TSIRM for ex45 of PETSc/KSP with two preconditioner (ASM and HYPRE) having 25,000 components per core on Curie ($\epsilon_{tsirm}=1e-10$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}\r
-\label{tab:06}\r
-\end{center}\r
-\end{table*}\r
-\r
-\r
\begin{figure}[htbp]\r
\centering\r
\includegraphics[width=0.5\textwidth]{nb_iter_sec_ex45_curie}\r
\end{figure}\r
\r
\r
+%%NEW\r
+\r
+\subsection{Parallel nonlinear problems}\r
+\r
+With PETSc, linear solvers are used inside nonlinear solvers. The SNES\r
+(Scalable Nonlinear Equations Solvers) module in PETSc implements easy to use\r
+methods, like Newton-type, quasi-Newton or full approximation scheme (FAS)\r
+multigrid to solve systems of nonlinears equations. As the SNES is based on the\r
+Krylov methods of PETSc, it is interesting to investigate if the TSIRM method is\r
+also efficient and scalable with non linear problems.\r
+\r
+\r
+\r
\r
\begin{table*}[htbp]\r
\begin{center}\r
\end{table*}\r
\r
\r
+\subsection{Influence of parameters for TSIRM}\r
+\r
+\r
+\r
+\r
+\r
+\subsection{Experiments conclusions }\r
+\r
+{\bf A refaire}\r
+\r
+Concerning the experiments some other remarks are interesting.\r
+\begin{itemize}\r
+\item We have tested other examples of PETSc/KSP (ex29, ex45, ex49). For all these\r
+ examples, we have also obtained similar gains between GMRES and TSIRM but\r
+ those examples are not scalable with many cores. In general, we had some\r
+ problems with more than $4,096$ cores.\r
+\item We have tested many iterative solvers available in PETSc. In fact, it is\r
+ possible to use most of them with TSIRM. From our point of view, the condition\r
+ to use a solver inside TSIRM is that the solver must have a restart\r
+ feature. More precisely, the solver must support to be stopped and restarted\r
+ without decreasing its convergence. That is why with GMRES we stop it when it\r
+ is naturally restarted (\emph{i.e.} with $m$ the restart parameter). The\r
+ Conjugate Gradient (CG) and all its variants do not have ``restarted'' version\r
+ in PETSc, so they are not efficient. They will converge with TSIRM but not\r
+ quickly because if we compare a normal CG with a CG which is stopped and\r
+ restarted every 16 iterations (for example), the normal CG will be far more\r
+ efficient. Some restarted CG or CG variant versions exist and may be\r
+ interesting to study in future works.\r
+\end{itemize}\r
+\r
%%ENDNEW\r
\r
+\r
%%%*********************************************************\r
%%%*********************************************************\r
\section{Conclusion}\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
-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 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
+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
-% conference papers do not normally have an appendix\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
+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
