X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/acead508c1f820b919aa203c78668e5e645cc9b8..59bde6ffdfbc419978262601f80b4a376f26d4d0:/paper.tex diff --git a/paper.tex b/paper.tex index bf9e767..6ae2692 100644 --- a/paper.tex +++ b/paper.tex @@ -380,7 +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\\ @@ -669,8 +669,8 @@ called for a maximum of $max\_iter_{kryl}$ iterations. In practice, we sugges equals to the restart number of the GMRES-like method. Moreover, a tolerance threshold must be specified for the solver. In practice, this threshold must be much smaller than the convergence threshold of the TSIRM algorithm (\emph{i.e.} -$\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k~ mod~ s}=x^k$ consists in copying the -solution $x_k$ into the column $k~ mod~ s$ of the matrix $S$. After the +$\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k \mod s}=x^k$ consists in copying the +solution $x_k$ into the column $k \mod s$ of the matrix $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. After the minimization, the matrix $S$ is reused with the new values of the residuals. To solve the minimization problem, an iterative method is used. Two parameters are required for that: the maximum number of iterations and the threshold to stop the @@ -686,13 +686,13 @@ Let us summarize the most important parameters of TSIRM: \end{itemize} -The parallelisation of TSIRM relies on the parallelization of all its +The parallelization of TSIRM relies on the parallelization of all its parts. More precisely, except the least-squares step, all the other parts are obvious to achieve out in parallel. In order to develop a parallel version of our code, we have chosen to use PETSc~\cite{petsc-web-page}. For line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and efficient since the matrix $A$ is sparse and since the matrix $S$ contains few -colums in practice. As explained previously, at least two methods seem to be +columns in practice. As explained previously, at least two methods seem to be interesting to solve the least-squares minimization, CGLS and LSQR. In the following we remind the CGLS algorithm. The LSQR method follows more or @@ -737,15 +737,53 @@ these operations are easy to implement in PETSc or similar environment. \label{sec:04} Let us recall the following result, see~\cite{Saad86}. \begin{proposition} +\label{prop:saad} Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies: \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} +We can now claim that, +\begin{proposition} +If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, we still have +\begin{equation} +||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| , +\end{equation} +where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}. +\end{proposition} + +\begin{proof} +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 by a mathematical induction that, for each $k \in \mathbb{N}^\ast$, +$||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0||.$ + +The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}. + +Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0||$. +We will show that the statement holds too for $r_k$. Two situations can occur: +\begin{itemize} +\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0||$. + +\item Else, let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\ +$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$ + +$\begin{array}{ll} +& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\ +& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\ +& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\ +& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\ +& \leqslant ||b-Ax_{k-1}||_2 . +\end{array}$ +\end{itemize} +\end{proof} + +We can remark that, at each iterate, the residue of the TSIRM algorithm is lower +than the one of the GMRES method. %%%********************************************************* %%%********************************************************* @@ -817,7 +855,7 @@ torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\ \hline \end{tabular} -\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.} +\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.} \label{tab:02} \end{center} \end{table} @@ -876,7 +914,7 @@ Table~\ref{tab:03} shows the execution times and the number of iterations of 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 +problems) per core 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 small. @@ -905,7 +943,7 @@ 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 +Table~\ref{tab:03} is displayed. It can be noticed that the number of 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, @@ -923,7 +961,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \begin{tabular}{|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & threshold & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ + nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ \cline{3-8} & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline 2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\ @@ -936,7 +974,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting. \hline \end{tabular} -\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.} \label{tab:04} \end{center} \end{table*} @@ -950,9 +988,9 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} \hline - nb. cores & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ + nb. cores & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ \cline{2-7} \cline{9-11} - & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & GMRES & TS CGLS & TS LSQR\\\hline \hline + & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & & FGMRES & TS CGLS & TS LSQR\\\hline \hline 512 & 3,969.69 & 33,120 & 709.57 & 5,790 & 622.76 & 5,070 & 6.37 & 1 & 1 & 1 \\ 1024 & 1,530.06 & 25,860 & 290.95 & 4,830 & 307.71 & 5,070 & 5.25 & 1.30 & 1.21 & 1.01 \\ 2048 & 919.62 & 31,470 & 237.52 & 8,040 & 194.22 & 6,510 & 4.73 & 1.08 & .75 & .80\\ @@ -962,7 +1000,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect \hline \end{tabular} -\caption{Comparison of FGMRES and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshol 5e-5), time is expressed in seconds.} +\caption{Comparison of FGMRES and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshold 5e-5), time is expressed in seconds.} \label{tab:05} \end{center} \end{table*} @@ -990,7 +1028,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect future plan : \\ - study other kinds of matrices, problems, inner solvers\\ -- test the influence of all the parameters\\ +- test the influence of all parameters\\ - adaptative number of outer iterations to minimize\\ - other methods to minimize the residuals?\\ - implement our solver inside PETSc @@ -1005,7 +1043,7 @@ future plan : \\ %%%********************************************************* \section*{Acknowledgment} This paper is partially funded by the Labex ACTION program (contract -ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resource +ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources Curie and Juqueen respectively based in France and Germany. @@ -1048,5 +1086,3 @@ Curie and Juqueen respectively based in France and Germany. % that's all folks \end{document} - -