X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/9c834284514e7988ab1e0780d1b58a9a9f317751..9deeb3b22421122e2872c4c432662811ec125909:/paper.tex diff --git a/paper.tex b/paper.tex index 087ff6e..e4421cd 100644 --- a/paper.tex +++ b/paper.tex @@ -353,6 +353,7 @@ \usepackage{algpseudocode} \usepackage{amsmath} \usepackage{amssymb} +\usepackage{multirow} \algnewcommand\algorithmicinput{\textbf{Input:}} \algnewcommand\Input{\item[\algorithmicinput]} @@ -366,31 +367,29 @@ % % paper title % can use linebreaks \\ within to get better formatting as desired -\title{A Krylov two-stage algorithm to solve large sparse linear systems} +\title{TSARM: A Two-Stage Algorithm with least-square Residual Minimization to solve large sparse linear systems} %où %\title{A two-stage algorithm with error minimization to solve large sparse linear systems} %où %\title{???} + + + % author names and affiliations % use a multiple column layout for up to two different % affiliations -\author{\IEEEauthorblockN{Rapha\"el Couturier} -\IEEEauthorblockA{Femto-ST Institute - DISC Department\\ -Universit\'e de Franche-Comt\'e, IUT de Belfort-Montb\'eliard\\ -19 avenue de Mar\'echal Juin, BP 527 \\ -90016 Belfort Cedex, France\\ -Email: raphael.couturier@univ-fcomte.fr} -\and -\IEEEauthorblockN{Lilia Ziane Khodja} -\IEEEauthorblockA{Centre de Recherche INRIA Bordeaux Sud-Ouest\\ -200 avenue de la Vieille Tour\\ -33405 Talence Cedex, France\\ +\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\\ Email: lilia.ziane@inria.fr} } + + % conference papers do not typically use \thanks and this command % is locked out in conference mode. If really needed, such as for % the acknowledgment of grants, issue a \IEEEoverridecommandlockouts @@ -427,7 +426,16 @@ Email: lilia.ziane@inria.fr} \begin{abstract} -%The abstract goes here. DO NOT USE SPECIAL CHARACTERS, SYMBOLS, OR MATH IN YOUR TITLE OR ABSTRACT. +In this paper we propose a two stage iterative method which increases the +convergence of Krylov iterative methods, typically those of GMRES variants. The +principle of our approach is to build an external iteration over the Krylov +method and to save the current residual frequently (for example, for each +restart of GMRES). Then after a given number of outer iterations, a minimization +step is applied on the matrix composed of the save residuals in order to compute +a better solution and make a new iteration if necessary. We prove that our +method has the same convergence property than the inner method used. Some +experiments using up to 16,394 cores show that compared to GMRES our algorithm +can be around 7 times faster. \end{abstract} \begin{IEEEkeywords} @@ -538,6 +546,7 @@ Iterative Krylov methods; sparse linear systems; error minimization; PETSc; %à % no \IEEEPARstart % You must have at least 2 lines in the paragraph with the drop letter % (should never be an issue) +{\bf RAPH : EST ce qu'on parle de Krylov pour dire que les résidus constituent une base de Krylov... J'hésite... Tof t'en penses quoi?} Iterative methods are become more attractive than direct ones to solve very large sparse linear systems. They are more effective in a parallel context and require less memory and arithmetic operations than direct methods. A number of @@ -636,8 +645,8 @@ direct method in the parallel context. \Input $A$ (sparse matrix), $b$ (right-hand side) \Output $x$ (solution vector)\vspace{0.2cm} \State Set the initial guess $x^0$ - \For {$k=1,2,3,\ldots$ until convergence} - \State Solve iteratively $Ax^k=b$ + \For {$k=1,2,3,\ldots$ until convergence} \label{algo:conv} + \State Solve iteratively $Ax^k=b$ \label{algo:solve} \State $S_{k~mod~s}=x^k$ \If {$k$ mod $s=0$ {\bf and} not convergence} \State Compute dense matrix $R=AS$ @@ -663,6 +672,127 @@ reused with the new values of the residuals. \section{Experiments using petsc} \label{sec:04} + +In order to see the influence of our algorithm with only one processor, we first +show a comparison with the standard version of GMRES and our algorithm. In +table~\ref{tab:01}, we show the matrices we have used and some of them +characteristics. For all the matrices, the name, the field, the number of rows +and the number of nonzero elements is given. + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|r|r|r|} +\hline +Matrix name & Field &\# Rows & \# Nonzeros \\\hline \hline +crashbasis & Optimization & 160,000 & 1,750,416 \\ +parabolic\_fem & Computational fluid dynamics & 525,825 & 2,100,225 \\ +epb3 & Thermal problem & 84,617 & 463,625 \\ +atmosmodj & Computational fluid dynamics & 1,270,432 & 8,814,880 \\ +bfwa398 & Electromagnetics problem & 398 & 3,678 \\ +torso3 & 2D/3D problem & 259,156 & 4,429,042 \\ +\hline + +\end{tabular} +\caption{Main characteristics of the sparse matrices chosen from the Davis collection} +\label{tab:01} +\end{center} +\end{table} + +The following parameters have been chosen for our experiments. As by default +the restart of GMRES is performed every 30 iterations, we have chosen to stop +the GMRES every 30 iterations (line \ref{algo:solve} in +Algorithm~\ref{algo:01}). $s$ is set to 8. CGLS is chosen to minimize the +least-squares problem. Two conditions are used to stop CGLS, either the +precision is under $1e-40$ or the number of iterations is greater to $20$. The +external precision is set to $1e-10$ (line \ref{algo:conv} in +Algorithm~\ref{algo:01}). 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. + + +\begin{table} +\begin{center} +\begin{tabular}{|c|c|r|r|r|r|} +\hline + + \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} \\ +\cline{3-6} + & precond & Time & \# Iter. & Time & \# Iter. \\\hline \hline + +crashbasis & gmres / none & 15.65 & 518 & 14.12 & 450 \\ +parabolic\_fem & gmres / ilu & 1009.94 & 7573 & 401.52 & 2970 \\ +epb3 & fgmres / sor & 8.67 & 600 & 8.21 & 540 \\ +atmosmodj & fgmres / sor & 104.23 & 451 & 88.97 & 366 \\ +bfwa398 & gmres / none & 1.42 & 9612 & 0.28 & 1650 \\ +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.} +\label{tab:02} +\end{center} +\end{table} + + + + +Larger experiments .... + +\begin{table*} +\begin{center} +\begin{tabular}{|r|r|r|r|r|r|r|r|r|} +\hline + + nb. cores & precond & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage 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 \\ + 2,048 & sor & 745.37 & 57,060 & 87.31 & 6,150 & 104.21 & 7,230 & 8.53 \\ + 4,096 & mg & 562.25 & 25,170 & 97.23 & 3,990 & 89.71 & 3,630 & 6.27 \\ + 4,096 & sor & 912.12 & 70,194 & 145.57 & 9,750 & 168.97 & 10,980 & 6.26 \\ + 8,192 & mg & 917.02 & 40,290 & 148.81 & 5,730 & 143.03 & 5,280 & 6.41 \\ + 8,192 & sor & 1,404.53 & 106,530 & 212.55 & 12,990 & 180.97 & 10,470 & 7.76 \\ + 16,384 & mg & 1,430.56 & 63,930 & 237.17 & 8,310 & 244.26 & 7,950 & 6.03 \\ + 16,384 & sor & 2,852.14 & 216,240 & 418.46 & 21,690 & 505.26 & 23,970 & 6.82 \\ +\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.} +\label{tab:03} +\end{center} +\end{table*} + + +\begin{table*} +\begin{center} +\begin{tabular}{|r|r|r|r|r|r|r|r|r|} +\hline + + nb. cores & threshold & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage CGLS} & \multicolumn{2}{c|}{2 stage LSQR} & best gain \\ +\cline{3-8} + & & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline + 8,192 & 6e-5 & 149.54 & 17,280 & 28.68 & 3,810 & 29.05 & 3,990 & 5.21 \\ + 8,192 & 5e-5 & 792.11 & 109,590 & 76.83 & 10,470 & 65.20 & 9,030 & 12.14 \\ + 16,384 & 4e-5 & 718.61 & 86,400 & 98.98 & 10,830 & 131.86 & 14,790 & 7.26 \\ +\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.} +\label{tab:04} +\end{center} +\end{table*} %%%********************************************************* %%%********************************************************* @@ -677,6 +807,12 @@ reused with the new values of the residuals. %%%********************************************************* +future plan : \\ +- study other kinds of matrices, problems, inner solvers\\ +- adaptative number of outer iterations to minimize\\ +- other methods to minimize the residuals?\\ +- implement our solver inside PETSc + % conference papers do not normally have an appendix @@ -686,10 +822,10 @@ reused with the new values of the residuals. %%%********************************************************* %%%********************************************************* \section*{Acknowledgment} -%The authors would like to thank... -%more thanks here -%%%********************************************************* -%%%********************************************************* +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 +Curie and Juqueen respectively based in France and Germany. + % trigger a \newpage just before the given reference