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-{\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
-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.
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-
+\section{Convergence results}
+\label{sec:04}
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\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
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 \\
