\newcommand{\Prec}{\mathit{prec}}
\newcommand{\Ratio}{\mathit{Ratio}}
-%\usepackage{xspace}
-%\usepackage[textsize=footnotesize]{todonotes}
-%\newcommand{\LZK}[2][inline]{%
-%\todo[color=green!40,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
+\let\endchangemargin=\endlist
\title{A scalable multisplitting algorithm for solving large sparse linear systems}
\date{}
%proposed in~\cite{huang1993krylov}, the use of a minimization process can
%drastically improve the convergence.
-Traditional parallel iterative solvers are based on fine-grain computations that frequently require data exchanges between computing nodes and have global synchronizations that penalize the scalability. Particularly, they are more penalized on large scale architectures or on distributed platforms composed of distant clusters interconnected by a high-latency network. It is therefore imperative to develop coarse-grain based algorithms to reduce the communications in the parallel iterative solvers. Two possible solutions consists either in using asynchronous iterative methods~\cite{ref18} or to use multisplitting algorithms. In this paper, we will reconsider the use of a multisplitting method. In opposition to traditional multisplitting method that suffer from slow convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization process can drastically improve the convergence.
-
-The present paper is organized as follows. First in Section~\ref{sec:02} is given some related works and the main principle of multisplitting methods. Then, in Section~\ref{sec:03} is presented the algorithm of our Krylov multisplitting method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the parallel experiments on Hector architecture show the performances of the Krylov multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D Poisson problem.
+Traditional parallel iterative solvers are based on fine-grain computations that
+frequently require data exchanges between computing nodes and have global
+synchronizations that penalize the scalability. Particularly, they are more
+penalized on large scale architectures or on distributed platforms composed of
+distant clusters interconnected by a high-latency network. It is therefore
+imperative to develop coarse-grain based algorithms to reduce the communications
+in the parallel iterative solvers. Two possible solutions consists either in
+using asynchronous iterative methods~\cite{ref18} or to use multisplitting
+algorithms. In this paper, we will reconsider the use of a multisplitting
+method. In opposition to traditional multisplitting method that suffer from slow
+convergence, as proposed in~\cite{huang1993krylov}, the use of a minimization
+process can drastically improve the convergence.
+
+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~\ref{sec:03} is presented the algorithm of our Krylov multisplitting
+method based on inner-outer iterations. Finally, in Section~\ref{sec:04}, the
+parallel experiments on Hector architecture show the performances of the Krylov
+multisplitting algorithm compared to the classical GMRES algorithm to solve a 3D
+Poisson problem.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[htbp]
\begin{center}
+\begin{changemargin}{-1.4cm}{0cm}
+\begin{footnotesize}
+\begin{tabular}{|c|c||c|c|c||c|c|c||c|}
+\hline
+\multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} & \multicolumn{3}{c||}{GMRES} & \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
+ \cline{3-8}
+ & & Time (s) & nb Iter. & $\Delta$ & Time (s)& nb Iter. & $\Delta$ & \\
+\hline
+$468^3$ & 2,048 (2x1,024) & 299.7 & 41,028 & 5.02e-8 & 48.4 & 691(6,146) & 8.24e-08 & 6.19 \\
+\hline
+$590^3$ & 4,096 (2x2,048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
+\hline
+$743^3$ & 8,192 (2x4,096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
+\hline
+$743^3$ & 8,192 (4x2,048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
+\hline
+
+\end{tabular}
+\caption{Results}
+\label{tab1}
+\end{footnotesize}
+\end{changemargin}
+\end{center}
+\end{table}
+
+
+
+
+\begin{table}[htbp]
+\begin{center}
+\begin{changemargin}{-1.8cm}{0cm}
+\begin{small}
\begin{tabular}{|c|c||c|c|c||c|c|c||c|}
\hline
\multirow{2}{*}{Pb size}&\multirow{2}{*}{Nb. cores} & \multicolumn{3}{c||}{GMRES} & \multicolumn{3}{c||}{Krylov Multisplitting} & \multirow{2}{*}{Ratio}\\
\end{tabular}
\caption{Results}
\label{tab1}
+\end{small}
+\end{changemargin}
\end{center}
\end{table}
