X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/e31850a4ae1376cd865f5e3fb950c9bbd3bdeeb8..9deeb3b22421122e2872c4c432662811ec125909:/paper.tex

diff --git a/paper.tex b/paper.tex
index b755e4c..e4421cd 100644
--- a/paper.tex
+++ b/paper.tex
@@ -367,13 +367,16 @@
 %
 % 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
@@ -423,7 +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}
@@ -534,6 +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
@@ -762,7 +775,24 @@ Larger experiments ....
 \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*}
 %%%*********************************************************
 %%%*********************************************************
 
@@ -777,6 +807,12 @@ Larger experiments ....
 %%%*********************************************************
 
 
+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