\right.
\label{sec03:eq01}
\end{equation}
-where for $l\in\{1,\ldots,L\}$ $A_l$ is a rectangular block of size $n_l\times n$
+where for $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size $n_l\times n$
and $X_l$ and $B_l$ are sub-vectors of size $n_l$, such that $\sum_ln_l=n$. In this
case, we use a row-by-row splitting without overlapping in such a way that successive
rows of the sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster.
iterative method which gives good performances for solving sparse linear systems in
parallel on a cluster of processors.
+It should be noted that the convergence of the inner iterative solver for the different
+linear sub-systems~(\ref{sec03:eq03}) does not necessarily involve the convergence of the
+multisplitting method. It strongly depends on the properties of the sparse linear system
+to be solved and the computing environment~\cite{o1985multi,ref18}. Furthermore, the multisplitting
+of the linear system among several clusters of processors increases the spectral radius
+of the iteration matrix, thereby slowing the convergence. In this paper, we based on the
+work presented in~\cite{huang1993krylov} to increase the convergence and improve the
+scalability of the multisplitting methods.
+
+In order to accelerate the convergence, we implement the outer iteration of the multisplitting
+solver as a Krylov subspace method which minimizes some error function over a Krylov subspace~\cite{S96}.
+The Krylov space of the method that we used is spanned by a basis composed of the solutions issued from
+solving the $L$ splittings~(\ref{sec03:eq03})
+\begin{equation}
+\{x^1,x^2,\ldots,x^s\},~s\ll n,
+\label{sec03:eq04}
+\end{equation}
+where for $k\in\{1,\ldots,s\}$, $x^k=[X_1^k,\ldots,X_L^k]$ is a solution of the global linear
+system.
+%The advantage such a method is that the Krylov subspace does not need to be spanned by an orthogonal basis.
+The advantage of such a Krylov subspace is that we need neither an orthogonal basis nor synchronizations
+between the different clusters to generate this basis.
+
