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
\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. 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
+$v_l^k$ 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}
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}
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 case, the parallel GMRES method is used
-as an inner iteration method for solving the linear sub-systems~(\ref{sec03:eq03}).
-
-
+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.
