X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/GMRES2stage.git/blobdiff_plain/33840a9b66d39bd2d78f409b7e25a8339280e05a..0f544a712fbfaa8e36e2d89273b1ecf21085669c:/paper.tex

diff --git a/paper.tex b/paper.tex
index 08e7b8a..3a51e45 100644
--- a/paper.tex
+++ b/paper.tex
@@ -621,10 +621,11 @@ outer solver periodically applies a least-squares minimization  on the residuals
 At each outer iteration, the sparse linear system $Ax=b$ is partially 
 solved using only $m$
 iterations of an iterative method, this latter being initialized with the 
-best known approximation previously obtained. 
-GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
-inner solver. The current approximation of the Krylov method is then stored inside a matrix
-$S$ composed by the successive solutions that are computed during inner iterations.
+last obtained approximation. 
+GMRES method~\cite{Saad86}, or any of its variants, can potentially be used as
+inner solver. The current approximation of the Krylov method is then stored inside a $n \times s$ matrix
+$S$, which is composed by the $s$ last solutions that have been computed during 
+the inner iterations phase.
 
 At each $s$ iterations, the minimization step is applied in order to
 compute a new  solution $x$. For that, the previous  residuals of $Ax=b$ are computed by
@@ -1029,6 +1030,15 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 %%%*********************************************************
 %%%*********************************************************
 
+A novel two-stage iterative  algorithm has been proposed in this article,
+in order to accelerate the convergence Krylov iterative  methods.
+Our TSIRM proposal acts as a merger between Krylov based solvers and
+a least-squares minimization step.
+The convergence of the method has been proven in some situations, while 
+experiments up to 16,394 cores have been led to verify that TSIRM runs
+5 or  7 times  faster than GMRES.
+
+
 For future work, the authors' intention is to investigate 
 other kinds of matrices, problems, and inner solvers. The 
 influence of all parameters must be tested too, while