%
% paper title
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-\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{???}
+
+
+
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-\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}
}
+
+
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% 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}
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% (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.
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-
+\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
characteristics. For all the matrices, the name, the field, the number of rows
and the number of nonzero elements is given.
-\begin{table}
+\begin{table*}
\begin{center}
\begin{tabular}{|c|c|r|r|r|}
\hline
\caption{Main characteristics of the sparse matrices chosen from the Davis collection}
\label{tab:01}
\end{center}
-\end{table}
+\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
\begin{tabular}{|c|c|r|r|r|r|}
\hline
- \multirow{2}{*}{Matrix name} & Solver / & \multicolumn{2}{c|}{gmres variant} & \multicolumn{2}{c|}{2 stage} \\
+ \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
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 \\
+torso3 & fgmres / sor & 37.70 & 565 & 34.97 & 510 \\
\hline
\end{tabular}
-\caption{Comparison of GMRES and 2 stage GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\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*}
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\section{Conclusion}
-\label{sec:05}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
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+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
+
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\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.
+
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