Relecture

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- 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 
 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
 
 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
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+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 
 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