-In order to accelerate the convergence, the outer iteration is implemented as an
-iterative Krylov method which minimizes some error functions over a Krylov
-subspace~\cite{saad96}. At each iteration, the sparse linear system $Ax=b$ is
-solved iteratively with an iterative method, for example GMRES
-method~\cite{saad86} or some of its variants, and the Krylov subspace that we
-used is spanned by a basis $S$ composed of successive solutions issued from the
-inner iteration
-\begin{equation}
- S = \{x^1, x^2, \ldots, x^s\} \text{,~} s\leq n.
-\end{equation}
-The advantage of such a Krylov subspace is that we neither need an orthogonal
-basis nor any synchronization between processors to generate this basis. The
-algorithm is periodically restarted every $s$ iterations with a new initial
-guess $x=S\alpha$ which minimizes the residual norm $\|b-Ax\|_2$ over the Krylov
-subspace spanned by vectors of $S$, where $\alpha$ is a solution of the normal
-equations
-\begin{equation}
- R^TR\alpha = R^Tb,
-\end{equation}
-which is associated with the least-squares problem
+In order to accelerate the convergence, the outer iteration periodically applies
+a least-square minimization on the residuals computed by the inner solver. The
+inner solver is a Krylov based solver which does not required to be changed.
+
+At each outer iteration, the sparse linear system $Ax=b$ is solved, only for $m$
+iterations, using an iterative method restarting with the previous solution. For
+example, the GMRES method~\cite{Saad86} or some of its variants can be used as a
+inner solver. The current solution of the Krylov method is saved inside a matrix
+$S$ composed of successive solutions computed by the inner iteration.
+
+Periodically, every $s$ iterations, the minimization step is applied in order to
+compute a new solution $x$. For that, the previous residuals are computed with
+$(b-AS)$. The minimization of the residuals is obtained by
