\usepackage{graphicx}
\usepackage{algorithm}
\usepackage{algpseudocode}
+\usepackage{multirow}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
\algnewcommand\Output{\item[\algorithmicoutput]}
+\newcommand{\Time}[1]{\mathit{Time}_\mathit{#1}}
+\newcommand{\Prec}{\mathit{prec}}
+\newcommand{\Ratio}{\mathit{Ratio}}
\title{A scalable multisplitting algorithm for solving large sparse linear systems}
+\date{}
\begin{abstract}
-In this paper we revist the krylov multisplitting algorithm presented in
+In this paper we revisit the krylov multisplitting algorithm presented in
\cite{huang1993krylov} which uses a scalar method to minimize the krylov
iterations computed by a multisplitting algorithm. Our new algorithm is based on
a parallel multisplitting algorithm with few blocks of large size using a
subroutines.
+\section{Experiments}
+
+In order to illustrate the interest of our algorithm. We have compared our
+algorithm with the GMRES method which a very well used method in many
+situations. We have chosen to focus on only one problem which is very simple to
+implement: a 3 dimension Poisson problem.
+
+\begin{equation}
+\left\{
+ \begin{array}{ll}
+ \nabla u&=f \mbox{~in~} \omega\\
+ u &=0 \mbox{~on~} \Gamma=\partial \omega
+ \end{array}
+ \right.
+\end{equation}
+
+After discretization, with a finite difference scheme, a seven point stencil is
+used. It is well-known that the spectral radius of matrices representing such
+problems are very close to 1. Moreover, the larger the number of discretization
+points is, the closer to 1 the spectral radius is. Hence, to solve a matrix
+obtained for a 3D Poisson problem, the number of iterations is high. Using a
+preconditioner it is possible to reduce the number of iterations but
+preconditioners are not scalable when using many cores.
+
+Doing many experiments with many cores is not easy and require to access to a
+supercomputer with several hours for developping a code and then improving
+it. In the following we presented some experiments we could achieved out on the
+Hector architecture, the previous UK's high-end computing resource, funded by
+the UK Research Councils, which has been stopped in the early 2014.
+
+In the experiments we report the size of the 3D poisson considered
+
+
+The first column shows the size of the problem The size is chosen in order to
+have approximately 50,000 components per core. The second column represents the
+number of cores used. In parenthesis, there is the decomposition used for the
+Krylov multisplitting. The third column and the sixth column respectively show
+the execution time for the GMRES and the Kyrlow multisplitting code. The fourth
+and the seventh column describes the number of iterations. For the
+multisplitting code, the total number of inner iterations is represented in
+parenthesis.
+
+ We also give the other parameters: the restart for the GRMES method....
+
+\begin{table}[p]
+\begin{center}
+\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
+
+$590^3$ & 4096 (2x2048) & 433.1 & 55,494 & 4.92e-7 & 74.1 & 1,101(8,211) & 6.62e-08 & 5.84 \\
+\hline
+$743^3$ & 8192 (2x4096) & 704.4 & 87,822 & 4.80e-07 & 151.2 & 3,061(14,914) & 5.87e-08 & 4.65 \\
+\hline
+$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 110.3 & 1,531(12,721) & 1.47e-07& 6.39 \\
+\hline
+
+\end{tabular}
+\caption{Results without preconditioner}
+\label{tab1}
+\end{center}
+\end{table}
+
+
+\begin{table}[p]
+\begin{center}
+\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
+
+$590^3$ & 4096 (2x2048) & 433.0 & 55,494 & 4.92e-7 & 80.4 & 1,091(9,545) & 7.64e-08 & 5.39 \\
+\hline
+$743^3$ & 8192 (2x4096) & 704.4 & 87,822 & 4.80e-07 & 110.2 & 1,401(12,379) & 1.11e-07 & 6.39 \\
+\hline
+$743^3$ & 8192 (4x2048) & 704.4 & 87,822 & 4.80e-07 & 139.8 & 1,891(15,960) & 1.60e-07& 5.03 \\
+\hline
+
+\end{tabular}
+\caption{Results with preconditioner}
+\label{tab2}
+\end{center}
+\end{table}
+
+\section{Conclusion and perspectives}
+
+Other applications (=> other matrices)\\
+Larger experiments\\
+Async\\
+Overlapping
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