Iterative methods are used to solve large sparse linear systems of equations of
the form $Ax=b$ because they are easier to parallelize than direct ones. Many
-iterative methods have been proposed and adapted by many researchers. When
-solving large linear systems with many cores, iterative methods often suffer
-from scalability problems. This is due to their need for collective
+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
The advantage of the multisplitting method is that at each iteration
$k$ there are $L$ different linear sub-systems
\begin{equation}
-y_l^k=M^{-1}_l N_l x_l^{k-1} + M^{-1}_l b,~l\in\{1,\ldots,L\},
+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
-$y_l^k$ is the solution of the local sub-system. 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 $y_l^k$ 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
+$v_l^k$ 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 y_l^k,
+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. $y_l^k$ are disjoint vectors), the
+and one factors (i.e. $v_l^k$ are disjoint vectors), the
multisplitting method is non-overlapping and corresponds to the block
Jacobi method.
%%%%%%%%%%%%%%%%%%%%%%%
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 a asynchronous version \cite{bru1995parallel} and convergence
-condition \cite{bai1999block,bahi2000asynchronous} in this case or other
-two-stage algorithms~\cite{frommer1992h,bru1995parallel}
+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 tasks minimize the error function over the basis to
+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.
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.
%%%%%%%%%%%%%%%%%%%%%%%%
\right.
\label{sec03:eq01}
\end{equation}
-where for all $l\in\{1,\ldots,L\}$ $A_l$ 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.
\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 $i\in\{1,\ldots,l-1,l+1,\ldots,L\}$. Therefore, each cluster $l$ is in charge of solving
-the following spare sub-linear system:
+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}
\right.
\label{sec03:eq03}
\end{equation}
-where the sub-vectors $X_i$ define the data dependencies between the cluster $l$ and other clusters.
+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.
+
+
+
