-\documentclass[conference]{IEEEtran}
+\documentclass[times]{cpeauth}
+
+\usepackage{moreverb}
+
+%\usepackage[dvips,colorlinks,bookmarksopen,bookmarksnumbered,citecolor=red,urlcolor=red]{hyperref}
+
+%\newcommand\BibTeX{{\rmfamily B\kern-.05em \textsc{i\kern-.025em b}\kern-.08em
+%T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+
+\def\volumeyear{2015}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{xspace}
\usepackage[textsize=footnotesize]{todonotes}
+
\newcommand{\AG}[2][inline]{%
\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
\newcommand{\RC}[2][inline]{%
\definecolor{Gray}{gray}{0.9}
+
\begin{document}
\RCE{Titre a confirmer.}
-
\title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+%\itshape{\journalnamelc}\footnotemark[2]}
-\author{%
- \IEEEauthorblockN{%
- Charles Emile Ramamonjisoa and
+\author{ Charles Emile Ramamonjisoa and
David Laiymani and
Arnaud Giersch and
Lilia Ziane Khodja and
Raphaël Couturier
- }
- \IEEEauthorblockA{%
+}
+
+\address{
+ \centering
Femto-ST Institute - DISC Department\\
Université de Franche-Comté\\
Belfort\\
Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
- }
}
-\maketitle
-
\begin{abstract}
ABSTRACT
+\end{abstract}
+\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
-Keywords : Algorithm distributed iterative asynchronous simulation simgrid performance
-
-\end{abstract}
+\maketitle
\section{Introduction}
\section{SimGrid}
-\section{Simulation of the multisplitting method}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage splitting methods}
+\label{sec:04}
+\subsection{Multisplitting methods for sparse linear systems}
+\label{sec:04.01}
+Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:
+\begin{equation}
+Ax=b,
+\label{eq:01}
+\end{equation}
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows:
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+\label{eq:02}
+\end{equation}
+where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix.
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental, Results and Comments}
\textit{3.a Executing the algorithms on various computational grid
architecture scaling up the input matrix size}
-
+\\
% environment
\begin{footnotesize}
Table 1 : Clusters x Nodes with NX=150 or NX=170
-\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-\begin{wrapfigure}{l}{50mm}
-\centering
-\includegraphics[width=50mm]{Cluster x Nodes NX=150 and NX=170.jpg}
-\caption{Cluster x Nodes NX=150 and NX=170 \label{overflow}}
-\end{wrapfigure}
+\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
The results in figure 1 show the non-variation of the number of
iterations of classical GMRES for a given input matrix size; it is not
-the case for the multisplitting method. Unless the 8x8 cluster, the time
+the case for the multisplitting method.
+
+%\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\caption{Cluster x Nodes NX=150 and NX=170}
+%\label{overflow}}
+\end{figure}
+%\end{wrapfigure}
+
+Unless the 8x8 cluster, the time
execution difference between the two algorithms is important when
comparing between different grid architectures, even with the same number of
processors (like 2x16 and 4x8 = 32 processors for example). The
Grid & 2x16, 4x8\\ %\hline
Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
- & N2 : bw=1Gbs-lat=5E-05 \\
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
\end{tabular}
\end{footnotesize}
%\RCE{idem pour tous les tableaux de donnees}
-\begin{wrapfigure}{l}{45mm}
+%\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
\centering
-\includegraphics[width=50mm]{Cluster x Nodes N1 x N2.jpg}
-\caption{Cluster x Nodes N1 x N2\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf}
+\caption{Cluster x Nodes N1 x N2}
+%\label{overflow}}
+\end{figure}
+%\end{wrapfigure}
The experiments compare the behavior of the algorithms running first on
speed inter- cluster network (N1) and a less performant network (N2).
when the network speed drops down, the difference between the execution
times can reach more than 25\%.
-\textit{3.c Network latency impacts on performance}
+\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance}
% environment
\begin{footnotesize}
\hline
Grid & 2x16\\ %\hline
Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
\end{tabular}
\end{footnotesize}
Table 3 : Network latency impact
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
\centering
-\includegraphics[width=60mm]{Network latency impact on execution time.jpg}
-\caption{Network latency impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impact on execution time}
+%\label{overflow}}
+\end{figure}
According the results in table and figure 3, degradation of the network
Table 4 : Network bandwidth impact
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
\centering
-\includegraphics[width=60mm]{Network bandwith impact on execution time.jpg}
-\caption{Network bandwith impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impact on execution time}
+%\label{overflow}
+\end{figure}
Table 5 : Input matrix size impact
-\begin{wrapfigure}{l}{50mm}
+\begin{figure} [ht!]
\centering
-\includegraphics[width=60mm]{Pb size impact on execution time.jpg}
-\caption{Pb size impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Pb size impact on execution time}
+%\label{overflow}}
+\end{figure}
In this experimentation, the input matrix size has been set from
Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to
Table 6 : CPU Power impact
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
\centering
-\includegraphics[width=60mm]{CPU Power impact on execution time.jpg}
-\caption{CPU Power impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf}
+\caption{CPU Power impact on execution time}
+%\label{overflow}}
+\end{figure}
Using the SIMGRID simulator flexibility, we have tried to determine the
impact on the algorithms performance in varying the CPU power of the
