+In~\cite{prace-multi}, the authors have proposed a parallel multisplitting
+algorithm in which large block are solved using a GMRES solver. The authors have
+performed large scale experimentations upto 32.768 cores and they conclude that
+asynchronous multisplitting algorithm could more efficient than traditionnal
+solvers on exascale architecture with hunders of thousands of cores.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\section{A two-stage method with a minimization}
+Let $Ax=b$ be a given sparse and large linear system of $n$ equations
+to solve in parallel on $L$ clusters, physically adjacent or geographically
+distant, where $A\in\mathbb{R}^{n\times n}$ is a square and nonsingular
+matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$
+is the right-hand side vector. The multisplitting of this linear system
+is defined as follows:
+\begin{equation}
+\left\{
+\begin{array}{lll}
+A & = & [A_{1}, \ldots, A_{L}]\\
+x & = & [X_{1}, \ldots, X_{L}]\\
+b & = & [B_{1}, \ldots, B_{L}]
+\end{array}
+\right.
+\label{sec03:eq01}
+\end{equation}
+where for $l\in\{1,\ldots,L\}$, $A_l$ is a rectangular block of size $n_l\times n$
+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.
+So, the multisplitting format of the linear system is defined as follows:
+\begin{equation}
+\forall l\in\{1,\ldots,L\} \mbox{,~} \displaystyle\sum_{i=1}^{l-1}A_{li}X_i + A_{ll}X_l + \displaystyle\sum_{i=l+1}^{L}A_{li}X_i = B_l,
+\label{sec03:eq02}
+\end{equation}
+where $A_{li}$ is a block of size $n_l\times n_i$ of the rectangular matrix $A_l$, $X_i\neq X_l$
+is a sub-vector of size $n_i$ of the solution vector $x$ and $\sum_{i<l}n_i+\sum_{i>l}n_i+n_l=n$,
+for all $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$.
+
+The multisplitting method proceeds by iteration for solving the linear system in such a
+way each sub-system
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{sec03:eq03}
+\end{equation}
+is solved independently by a cluster of processors and communication are required to
+update the right-hand side vectors $Y_l$, such that the vectors $X_i$ represent the data
+dependencies between the clusters. In this work, we use the GMRES method as an inner
+iteration method for solving the sub-systems~(\ref{sec03:eq03}). It is a well-known
+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 method is that the Krylov subspace need neither to be spanned by an orthogonal
+basis nor synchronizations between the different clusters to generate this basis.
+
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%
+
