\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:
+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}
\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:
+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 $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$.
+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
+The multisplitting method proceeds by iteration for solving the linear
+system in such a way each sub-system
\begin{equation}
\left\{
\begin{array}{l}
\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})
+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 successive solutions issued from solving the $L$
+splittings~(\ref{sec03:eq03})
\begin{equation}
-\{x^1,x^2,\ldots,x^s\},~s\ll n,
+S=\{x^1,x^2,\ldots,x^s\},~s\leq 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.
-
-
+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.
+
+The multisplitting method is periodically restarted every $s$ iterations
+with a new initial guess $\tilde{x}=S\alpha$ which minimizes the error
+function $\|b-Ax\|_2$ over the Krylov subspace spanned by the vectors of $S$.
+So, $\alpha$ is defined as the solution of the large overdetermined linear system
+\begin{equation}
+B\alpha=b,
+\label{sec03:eq05}
+\end{equation}
+where $B=AS$ is a dense rectangular matrix of size $n\times s$ and $s\ll n$. This leads
+us to solve the system of normal equations
+\begin{equation}
+B^TB\alpha=B^Tb,
+\label{sec03:eq06}
+\end{equation}
+which is associated with the least squares problem
+\begin{equation}
+\text{minimize}~\|b-B\alpha\|_2,
+\label{sec03:eq07}
+\end{equation}
+where $B^T$ denotes the transpose of the matrix $B$. Since $B$ (i.e. $AS$) and
+$b$ are split among $L$ clusters, the symmetric positive definite system~(\ref{sec03:eq06})
+is solved in parallel. Thus, an iterative method would be more appropriate than
+a direct one for solving this system. We use the parallel conjugate gradient
+method for the normal equations CGNR~\cite{S96,refCGNR}.
+
+%%% Ecrire l'algorithme(s)
+%%% Parler des synchronisations entre proc et clusters
