\usepackage{algpseudocode}
\usepackage{amsmath}
\usepackage{amssymb}
+\usepackage{multirow}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
%
% 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
\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}
% no \IEEEPARstart
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
-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
-iterative methods are proposed and adapted by many researchers and the increased
-need for solving very large sparse linear systems triggered the development of
-efficient iterative techniques suitable for the parallel processing.
+
+Iterative methods became more attractive than direct ones to solve very large
+sparse linear systems. Iterative methods are more effecient in a parallel
+context, with thousands of cores, and require less memory and arithmetic
+operations than direct methods. A number of iterative methods are proposed and
+adapted by many researchers and the increased need for solving very large sparse
+linear systems triggered the development of efficient iterative techniques
+suitable for the parallel processing.
Most of the successful iterative methods currently available are based on Krylov
subspaces which consist in forming a basis of a sequence of successive matrix
In this paper we propose a two-stage algorithm based on two nested iterations
called inner-outer iterations. This algorithm consists in solving the sparse
linear system iteratively with a small number of inner iterations and restarts
-the outer step with a new solution minimizing some error functions over a Krylov
-subspace. This algorithm is iterative and easy to parallelize on large clusters
-and the minimization technique improves its convergence and performances.
+the outer step with a new solution minimizing some error functions over some
+previous residuals. This algorithm is iterative and easy to parallelize on large
+clusters and the minimization technique improves its convergence and
+performances.
The present paper is organized as follows. In Section~\ref{sec:02} some related
-works are presented. Section~\ref{sec:03} presents our two-stage algorithm based
-on Krylov subspace iteration methods. Section~\ref{sec:04} shows some
+works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
+a least-square residual minimization. Section~\ref{sec:04} describes some
+convergence results on this method. Section~\ref{sec:05} shows some
experimental results obtained on large clusters of our algorithm using routines
-of PETSc toolkit.
+of PETSc toolkit. Finally Section~\ref{sec:06} concludes and gives some
+perspectives.
%%%*********************************************************
%%%*********************************************************
\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 $R=AS$
%%%*********************************************************
%%%*********************************************************
-
+\section{Convergence results}
+\label{sec:04}
%%%*********************************************************
%%%*********************************************************
\section{Experiments using petsc}
-\label{sec:04}
+\label{sec:05}
+
+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
+ 2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
+ 2,048 & 6e-5 & 194.01 & 30,270 & 35.50 & 5,430 & 27.74 & 4,350 & 6.99 \\
+ 4,096 & 7e-5 & 160.59 & 22,530 & 35.15 & 5,130 & 29.21 & 4,350 & 5.49 \\
+ 4,096 & 6e-5 & 249.27 & 35,520 & 52.13 & 7,950 & 39.24 & 5,790 & 6.35 \\
+ 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*}
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
\section{Conclusion}
-\label{sec:05}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
%%%*********************************************************
%%%*********************************************************
+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
%%%*********************************************************
%%%*********************************************************
\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
