X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/95fea275e81cda394a2278d66259fce8f4df5e8c..8aeef74d04b37c2601749676e053a538ba3785cd:/paper.tex diff --git a/paper.tex b/paper.tex index e927821..f1becbd 100644 --- a/paper.tex +++ b/paper.tex @@ -364,7 +364,6 @@ \algnewcommand\Output{\item[\algorithmicoutput]} \newtheorem{proposition}{Proposition} -\newtheorem{proof}{Proof} \begin{document} % @@ -381,7 +380,7 @@ % use a multiple column layout for up to two different % affiliations -\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} +\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}} \IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche Comte, France\\ Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr} \IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\ @@ -742,13 +741,16 @@ Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the res \begin{equation} ||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| , \end{equation} -where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves +where $\alpha = \lambda_min(M)^2$ and $\beta = \lambda_max(A^T A)$, which proves the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$. \end{proposition} +<<<<<<< HEAD + +======= We can now claim that, \begin{proposition} -If $A$ is a positive real matrix, then the TSIRM algorithm is convergent. +If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. \end{proposition} \begin{proof} @@ -756,8 +758,9 @@ Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the $k$-th iterate of TSIRM. We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$. - +Each step of the TSIRM algorithm \end{proof} +>>>>>>> 84e15020344b77e5497c4a516cc20b472b2914cd %%%********************************************************* %%%********************************************************* @@ -858,7 +861,7 @@ chosen because they are scalable with many cores which is not the case of other In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\ -{\bf Description of preconditioners} +{\bf Description of preconditioners}\\ \begin{table*}[htbp] \begin{center} @@ -879,15 +882,15 @@ In the following larger experiments are described on two large scale architectur \hline \end{tabular} -\caption{Comparison of FGMRES and TSIRM 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.} +\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (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 +example ex15 of PETSc on the Juqueen architecture. Different numbers of cores +are studied ranging from 2,048 up-to 16,383. Two preconditioners have been +tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the problems) per processor is fixed to 25,000, also called weak scaling. 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 @@ -895,11 +898,11 @@ small. -In this Table, we can notice that TSIRM is always faster than FGMRES. The last +In Table~\ref{tab:03}, we can notice that TSIRM is always faster than FGMRES. The last column shows the ratio between FGMRES and the best version of TSIRM according to the minimization procedure: CGLS or LSQR. Even if we have computed the worst -case between CGLS and LSQR, it is clear that TSIRM is alsways faster than -FGMRES. For this example, the multigrid preconditionner is faster than SOR. The +case between CGLS and LSQR, it is clear that TSIRM is always faster than +FGMRES. For this example, the multigrid preconditioner is faster than SOR. The gain between TSIRM and FGMRES is more or less similar for the two preconditioners. Looking at the number of iterations to reach the convergence, it is obvious that TSIRM allows the reduction of the number of iterations. It @@ -918,9 +921,9 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15. In Figure~\ref{fig:01}, the number of iterations per second corresponding to Table~\ref{tab:01} is displayed. It can be noticed that the number of -iterations per second of FMGRES is constant whereas it decrease with TSIRM with -both preconditioner. This can be explained by the fact that when the number of -core increases the time for the minimization step also increases but, generally, +iterations per second of FMGRES is constant whereas it decreases with TSIRM with +both preconditioners. This can be explained by the fact that when the number of +cores increases the time for the least-squares minimization step also increases but, generally, when the number of cores increases, the number of iterations to reach the threshold also increases, and, in that case, TSIRM is more efficient to reduce the number of iterations. So, the overall benefit of using TSIRM is interesting. @@ -1060,3 +1063,5 @@ Curie and Juqueen respectively based in France and Germany. % that's all folks \end{document} + +