+iterative methods have been proposed and adapted by many researchers. For
+example, the GMRES method and the Conjugate Gradient method are very well known
+and used by many researchers ~\cite{S96}. Both the method are based on the
+Krylov subspace which consists in forming a basis of the sequence of successive
+matrix powers times the initial residual.
+
+When solving large linear systems with many cores, iterative methods often
+suffer from scalability problems. This is due to their need for collective
+communications to perform matrix-vector products and reduction operations.
+Preconditionners can be used in order to increase the convergence of iterative
+solvers. However, most of the good preconditionners are not sclalable when
+thousands of cores are used.
+
+
+A completer...
+On ne peut pas parler de tout...\\
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+%% BEGIN
+%%%%%%%%%%%%%%%%%%%%%%%
+The key idea of the multisplitting method for solving a large system
+of linear equations $Ax=b$ consists in partitioning the matrix $A$ in
+$L$ several ways
+\begin{equation}
+A = M_l - N_l,~l\in\{1,\ldots,L\},
+\label{eq01}
+\end{equation}
+where $M_l$ are nonsingular matrices. Then the linear system is solved
+by iteration based on the multisplittings as follows
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{l=1} E_l M^{-1}_l (N_l x^k + b),~k=1,2,3,\ldots
+\label{eq02}
+\end{equation}
+where $E_l$ are non-negative and diagonal weighting matrices such that
+$\sum^L_{l=1}E_l=I$ ($I$ is an identity matrix). Thus the convergence
+of such a method is dependent on the condition
+\begin{equation}
+\rho(\displaystyle\sum^L_{l=1}E_l M^{-1}_l N_l)<1.
+\label{eq03}
+\end{equation}
+
+The advantage of the multisplitting method is that at each iteration
+$k$ there are $L$ different linear sub-systems
+\begin{equation}
+v_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+\label{eq04}
+\end{equation}
+to be solved independently by a direct or an iterative method, where
+$v_l^k$ is the solution of the local sub-system. Thus, the
+calculations of $v_l^k$ may be performed in parallel by a set of
+processors. A multisplitting method using an iterative method for
+solving the $L$ linear sub-systems is called an inner-outer iterative
+method or a two-stage method. The results $v_l^k$ obtained from the
+different splittings~(\ref{eq04}) are combined to compute the solution
+$x^k$ of the linear system by using the diagonal weighting matrices
+\begin{equation}
+x^k = \displaystyle\sum^L_{l=1} E_l v_l^k,
+\label{eq05}
+\end{equation}
+In the case where the diagonal weighting matrices $E_l$ have only zero
+and one factors (i.e. $v_l^k$ are disjoint vectors), the
+multisplitting method is non-overlapping and corresponds to the block
+Jacobi method.
+%%%%%%%%%%%%%%%%%%%%%%%
+%% END
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Related works}
+
+
+A general framework for studying parallel multisplitting has been presented in
+\cite{o1985multi} by O'Leary and White. Convergence conditions are given for the
+most general case. Many authors improved multisplitting algorithms by proposing
+for example an asynchronous version \cite{bru1995parallel} and convergence
+conditions \cite{bai1999block,bahi2000asynchronous} in this case or other
+two-stage algorithms~\cite{frommer1992h,bru1995parallel}.
+
+In \cite{huang1993krylov}, the authors proposed a parallel multisplitting
+algorithm in which all the tasks except one are devoted to solve a sub-block of
+the splitting and to send their local solution to the first task which is in
+charge to combine the vectors at each iteration. These vectors form a Krylov
+basis for which the first task minimizes the error function over the basis to
+increase the convergence, then the other tasks receive the update solution until
+convergence of the global system.
+
+
+
+In \cite{couturier2008gremlins}, the authors proposed practical implementations
+of multisplitting algorithms that take benefit from multisplitting algorithms to
+solve large scale linear systems. Inner solvers could be based on scalar direct
+method with the LU method or scalar iterative one with GMRES.
+
+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.