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
+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 matrix
$S$ composed by the successive solutions that are computed during inner iterations.
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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
