From fc8e999ae15f3227ad6bfa96711b70236c809e77 Mon Sep 17 00:00:00 2001 From: raphael couturier Date: Mon, 6 Oct 2014 17:07:52 +0200 Subject: [PATCH 1/1] new results --- paper.tex | 37 +++++++++++++++++++++++-------------- 1 file changed, 23 insertions(+), 14 deletions(-) diff --git a/paper.tex b/paper.tex index e4421cd..c559dda 100644 --- a/paper.tex +++ b/paper.tex @@ -546,13 +546,14 @@ 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 -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 @@ -574,13 +575,16 @@ large clusters. 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. %%%********************************************************* @@ -665,12 +669,13 @@ reused with the new values of the residuals. %%%********************************************************* %%%********************************************************* - +\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 @@ -783,6 +788,10 @@ Larger experiments .... 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 \\ -- 2.39.5