-\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]{%
\algnewcommand\algorithmicoutput{\textbf{Output:}}
\algnewcommand\Output{\item[\algorithmicoutput]}
-\newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\TOLG}{\mathit{tol_{gmres}}}
+\newcommand{\MIG}{\mathit{maxit_{gmres}}}
\usepackage{array}
\usepackage{color, colortbl}
\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
+%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
\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 $A$~\cite{O'leary85,White86}, 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. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system
+\begin{equation}
+M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b,
+\label{eq:03}
+\end{equation}
+then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
+\begin{equation}
+x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell.
+\label{eq:04}
+\end{equation}
+The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition
+%\begin{equation}
+%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
+%\label{eq:05}
+%\end{equation}
+%where $\rho$ is the spectral radius of the square matrix.
+The multisplitting methods are convergent:
+\begin{itemize}
+\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or
+\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous.
+\end{itemize}
+The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method).
+
+In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form
+\begin{equation}
+A_{\ell\ell} x_\ell^{k+1} = b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m},\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
+\label{eq:05}
+\end{equation}
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting
+\begin{equation}
+A_{\ell\ell} x_\ell = c_\ell,
+\label{eq:06}
+\end{equation}
+is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\ref{alg:01} shows the main key points of the block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of iterations and the tolerance threshold respectively.
+
+\begin{algorithm}[t]
+\caption{Block Jacobi two-stage method}
+\begin{algorithmic}[1]
+ \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+ \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+ \State Set the initial guess $x^0$
+ \For {$k=1,2,3,\ldots$ until convergence}
+ \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+ \State $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$ \label{solve}
+ \State Send $x_\ell^k$ to neighboring clusters
+ \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
+ \EndFor
+\end{algorithmic}
+\label{alg:01}
+\end{algorithm}
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental, Results and Comments}
\textbf{Step 2} : Collect the software materials needed for the
experimentation. In our case, we have three variants algorithms for the
-resolution of three 3D-Poisson problem: (1) using the classical GMRES
-\textit{(Generalized Minimal RESidual Method)} alias Algo-1 in this
+resolution of three 3D-Poisson problem: (1) using the classical GMRES alias Algo-1 in this
paper, (2) using the multisplitting method alias Algo-2 and (3) an
enhanced version of the multisplitting method as Algo-3. In addition,
SIMGRID simulator has been chosen to simulate the behaviors of the
\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
The authors would like to thank\dots{}
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-\bibliography{hpccBib}
+\bibliographystyle{wileyj}
+\bibliography{biblio}
\end{document}
