\usepackage{multirow}
\usepackage{authblk}
-\algnewcommand\algorithmicinput{\textbf{Input:}}
+
+\algnewcommand\algorithmicinput{\textbf{I1nput:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization
process can drastically improve the convergence.
+
+%%% AJOUTE************************
+%%%*******************************
In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to others iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.
+%%%*******************************
+%%%*******************************
The present paper is organized as follows. First, Section~\ref{sec:02} presents
some related works and the principle of multisplitting methods. Then, in
\section{A two-stage method with a minimization}
\label{sec:03}
-Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplittig method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+
+%%% MODIFIE ************************
+%%%*********************************
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ clusters of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+%%%*********************************
+%%%*********************************
+
+
\begin{equation}
\left\{
\begin{array}{lll}
\right.
\label{sec03:eq01}
\end{equation}
-where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. The splitting is performed row-by-row without overlapping in such a way that successive rows of 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 $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$.
+%%% MODIFIE ***********************
+%%%********************************
+The splitting is performed row-by-row without overlapping in such a way that successive rows of 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 \ell\in\{1,\ldots,L\} \mbox{,~} A_{\ell \ell}X_\ell + \displaystyle\sum_{\substack{m=1\\m\neq\ell}}^L A_{\ell m}X_m = B_\ell,
\label{sec03:eq02}
\right.
\label{sec03:eq03}
\end{equation}
-is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}). GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace. %In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
+is solved independently by a {\it cluster of processors} and communications are required to update the right-hand side vectors $Y_\ell$, such that the vectors $X_m$ represent the data dependencies between the clusters. In this work, we use the parallel restarted GMRES method~\cite{ref34} as an inner iteration method to solve sub-systems~(\ref{sec03:eq03}).
+%%% MODIFIE ***********************
+%%%********************************
+GMRES is one of the most used Krylov iterative methods to solve sparse linear systems by minimizing the residuals over an orthonormal basis of a Krylov subspace.
+%%%********************************
+%%%********************************
+%In practice, GMRES is used with a preconditioner to improve its convergence. In this work, we used a preconditioning matrix equivalent to the main diagonal of sparse sub-matrix $A_{ll}$. This preconditioner is straightforward to implement in parallel and gives good performances in many situations.
It should be noted that the convergence of the inner iterative solver for the
different sub-systems~(\ref{sec03:eq03}) does not necessarily involve the
work is based on the work presented in~\cite{huang1993krylov} to increase the
convergence and improve the scalability of the multisplitting methods.
-Krylov subspace methods are well-known for their good convergence compared to other iterative methods. In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
+%%% AJOUTE ************************
+%%%********************************
+Krylov subspace methods are well-known for their good convergence compared to other iterative methods.
+%%%********************************
+%%%********************************
+In order to accelerate the convergence, we implemented the outer iteration of our multisplitting solver as a Krylov iterative method which minimizes some error function over a Krylov subspace~\cite{S96}. The Krylov subspace that we used is spanned by a basis composed of successive solutions issued from solving the $L$ splittings~(\ref{sec03:eq03})
\begin{equation}
S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
\label{sec03:eq04}
\end{equation}
-where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system.
+%%% MODIFIE ***********************
+%%%********************************
+The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between clusters is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+%%%********************************
+%%%********************************
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 vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
\begin{equation}
\end{table}
+
+
+
From these experiments, it can be observed that the multisplitting version is
always faster than the GMRES version. The acceleration gain of the
multisplitting version ranges between 4 and 6. It can be noticed that the number of
precision with 2 clusters is slightly better but in both cases the precision is
under the specified threshold.
+
+%%% AJOUTE************************
+%%%*******************************
+In Figure~\ref{fig:01}, the number of iterations per second is reported for both
+GMRES and the multisplitting methods. It should be noted that we took only the
+inner number of iterations (i.e. the GMRES iterations) for the multisplitting
+method. Iterations of CGNR are not taken into account. From this figure, it can
+be seen that the number of iterations per second is higher with GMRES but it is
+not so different with the multisplitting method. For the case with $8,192$
+cores, the number of iterations per second with 4 clusters is approximately
+equals to 115. So it is not different from GMRES.
+
+
+\begin{figure}[htbp]
+\centering
+ \includegraphics[width=0.7\textwidth]{nb_iter_sec}
+\caption{Number of iterations per second with the same parameters as in Table~\ref{tab1} (weak scaling) with only 2 clusters}
+\label{fig:01}
+\end{figure}
+
+
+\noindent {\bf Final remarks:}\\
+It should be noted, on the one hand, that the development of a complete new
+method usable with any kind of problem is a really long and fastidious task if
+one is working from scratch. On the other hand, using an existing tool for the
+inner solver is also not easy because it requires to make link between the inner
+solver and the outer one. We plan to do that later with engineers working
+specifically on that point. Moreover, we think that it is very important to
+analyze the convergence of this method compared to other method. In this work,
+we have focused on the description of this method and its performance with a
+typical application. Many other investigations are required for this method as explained in the next section.
+%%%*******************************
+%%%*******************************
+
\section{Conclusion and perspectives}
We have implemented a Krylov multisplitting method to solve sparse linear
systems on large-scale computing platforms. We have developed a synchronous
