-
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-
-
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
-%\usepackage{amsmath}
+\usepackage{amsfonts,amssymb}
+\usepackage{amsmath}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
%\usepackage{amsthm}
-%\usepackage{amsfonts}
-%\usepackage{graphicx}
+\usepackage{graphicx}
%\usepackage{xspace}
\usepackage[american]{babel}
-% Extension pour les graphiques EPS
-%\usepackage[dvips]{graphicx}
-\usepackage[pdftex,final]{graphicx}
% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
-\begin{document}
-%
-% paper title
-% can use linebreaks \\ within to get better formatting as desired
-\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
-% author names and affiliations
-% use a multiple column layout for up to three different
-% affiliations
-\author{\IEEEauthorblockN{Raphaël Couturier and Arnaud Giersch and David Laiymani and Charles-Emile Ramamonjisoa}
-\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
-Université de Franche-Comté\\
-Belfort\\
-Email: raphael.couturier@univ-fcomte.fr}
-%\and
-%\IEEEauthorblockN{Arnaud Giersch}
-%\IEEEauthorblockA{Twentieth Century Fox\\
-%Springfield, USA\\
-%Email: homer@thesimpsons.com}
-%\and
-%\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-%\IEEEauthorblockA{Starfleet Academy\\
-%San Francisco, California 96678-2391\\
-%Telephone: (800) 555--1212\\
-%Fax: (888) 555--1212
-}
+\begin{document}
+\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\author{%
+ \IEEEauthorblockN{%
+ Raphaël Couturier,
+ Arnaud Giersch,
+ David Laiymani and
+ Charles Emile Ramamonjisoa
+ }
+ \IEEEauthorblockA{%
+ Femto-ST Institute - DISC Department\\
+ Université de Franche-Comté\\
+ Belfort\\
+ Email: raphael.couturier@univ-fcomte.fr
+ }
+}
-% make the title area
\maketitle
-
\begin{abstract}
-%\boldmath
The abstract goes here.
\end{abstract}
-% IEEEtran.cls defaults to using nonbold math in the Abstract.
-% This preserves the distinction between vectors and scalars. However,
-% if the conference you are submitting to favors bold math in the abstract,
-% then you can use LaTeX's standard command \boldmath at the very start
-% of the abstract to achieve this. Many IEEE journals/conferences frown on
-% math in the abstract anyway.
-% no keywords
+\section{Introduction}
+
+Parallel computing and high performance computing (HPC) are becoming
+more and more imperative for solving various problems raised by
+researchers on various scientific disciplines but also by industrial in
+the field. Indeed, the increasing complexity of these requested
+applications combined with a continuous increase of their sizes lead to
+write distributed and parallel algorithms requiring significant hardware
+resources ( grid computing , clusters, broadband network ,etc... ) but
+also a non- negligible CPU execution time. We consider in this paper a
+class of highly efficient parallel algorithms called iterative executed
+in a distributed environment. As their name suggests, these algorithm
+solves a given problem that might be NP- complete complex by successive
+iterations (X$_{n +1 }$= f (X$_{n}$) ) from an initial value X
+$_{0}$ to find an approximate value X* of the solution with a very low
+residual error. Several well-known methods demonstrate the convergence
+of these algorithms. Generally, to reduce the complexity and the
+execution time, the problem is divided into several "pieces" that will
+be solved in parallel on multiple processing units. The latter will
+communicate each intermediate results before a new iteration starts
+until the approximate solution is reached. These distributed parallel
+computations can be performed either in "synchronous" communication mode
+where a new iteration begin only when all nodes communications are
+completed, either "asynchronous" mode where processors can continue
+independently without or few synchronization points. Despite the
+effectiveness of iterative approach, a major drawback of the method is
+the requirement of huge resources in terms of computing capacity,
+storage and high speed communication network. Indeed, limited physical
+resources are blocking factors for large-scale deployment of parallel
+algorithms.
+
+In recent years, the use of a simulation environment to execute parallel
+iterative algorithms found some interests in reducing the highly cost of
+access to computing resources: (1) for the applications development life
+cycle and in code debugging (2) and in production to get results in a
+reasonable execution time with a simulated infrastructure not accessible
+with physical resources. Indeed, the launch of distributed iterative
+asynchronous algorithms to solve a given problem on a large-scale
+simulated environment challenges to find optimal configurations giving
+the best results with a lowest residual error and in the best of
+execution time. According our knowledge, no testing of large-scale
+simulation of the class of algorithm solving to achieve real results has
+been undertaken to date. We had in the scope of this work implemented a
+program for solving large non-symmetric linear system of equations by
+numerical method GMRES (Generalized Minimal Residual ) in the simulation
+environment Simgrid . The simulated platform had allowed us to launch
+the application from a modest computing infrastructure by simulating
+different distributed architectures composed by clusters nodes
+interconnected by variable speed networks. In addition, it has been
+permitted to show the effectiveness of asynchronous mode algorithm by
+comparing its performance with the synchronous mode time. With selected
+parameters on the network platforms (bandwidth, latency of inter cluster
+network) and on the clusters architecture (number, capacity calculation
+power) in the simulated environment , the experimental results have
+demonstrated not only the algorithm convergence within a reasonable time
+compared with the physical environment performance, but also a time
+saving of up to 40 \% in asynchronous mode.
+
+This article is structured as follows: after this introduction, the next
+section will give a brief description of iterative asynchronous model.
+Then, the simulation framework SIMGRID will be presented with the
+settings to create various distributed architectures. The algorithm of
+the multi -splitting method used by GMRES written with MPI primitives
+and its adaptation to Simgrid with SMPI (Simulation MPI ) will be in the
+next section . At last, the experiments results carried out will be
+presented before the conclusion which we will announce the opening of
+our future work after the results.
+
+\section{The asynchronous iteration model}
+
+Décrire le modèle asynchrone. Je m'en charge (DL)
+\section{SimGrid}
+Décrire SimGrid (Arnaud)
-% For peer review papers, you can put extra information on the cover
-% page as needed:
-% \ifCLASSOPTIONpeerreview
-% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
-% \fi
-%
-% For peerreview papers, this IEEEtran command inserts a page break and
-% creates the second title. It will be ignored for other modes.
-\IEEEpeerreviewmaketitle
-\section{Introduction}
-Présenter un bref état de l'art sur la simulation d'algos parallèles. Présenter rapidement les algos itératifs asynchrones et leurs avantages. Parler de leurs inconvénients en particulier la difficulté de déploiement à grande échelle donc il serait bien de simuler. Dire qu'à notre connaissance il n'existe pas de simulation de ce type d'algo.
-Présenter les travaux et les résultats obtenus. Annoncer le plan.
-
-\section{The asynchronous iteration model}
-Décrire le modèle asynchrone. Je m'en charge (DL)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Simulation of the multisplitting method}
+%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
+Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping
+\[
+\left(\begin{array}{ccc}
+A_{11} & \cdots & A_{1L} \\
+\vdots & \ddots & \vdots\\
+A_{L1} & \cdots & A_{LL}
+\end{array} \right)
+\times
+\left(\begin{array}{c}
+X_1 \\
+\vdots\\
+X_L
+\end{array} \right)
+=
+\left(\begin{array}{c}
+Y_1 \\
+\vdots\\
+Y_L
+\end{array} \right)\]
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.
+
+The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ clusters of processors, in such a way each sub-system
+\begin{equation}
+\left\{
+\begin{array}{l}
+A_{ll}X_l = Y_l \mbox{,~such that}\\
+Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+\end{array}
+\right.
+\label{eq:4.1}
+\end{equation}
+is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+
+\begin{algorithm}
+\caption{A multisplitting solver with inner iteration GMRES method}
+\begin{algorithmic}[1]
+\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
+\Output $X_l$ (local solution vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$, $x^0$
+\State Initialize the shared vector $\hat{x}=x^0$
+\For {$k=1,2,3,\ldots$ until the global convergence}
+\State $x^0=\hat{x}$
+\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
+\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
+\EndFor
+
+\Statex
+
+\Function {InnerSolver}{$x^0$, $k$}
+\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
+\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
+\State \Return $X_l^k$
+\EndFunction
+\end{algorithmic}
+\label{algo:01}
+\end{algorithm}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
-\section{SimGrid}
-Décrire SimGrid (Arnaud)
-\section{Simulation of the multi-splitting method}
-Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
\section{Experimental results}
\centering
\caption{2 clusters X 50 nodes}
\label{tab.cluster.2x50}
- \includegraphics[width=209pt]{img-1.eps}
+ \includegraphics[width=209pt]{img1.jpg}
\end{table}
\begin{table}
\centering
- \caption{3 clusters X 33 n\oe{}uds}
+ \caption{3 clusters X 33 nodes}
\label{tab.cluster.3x33}
- \includegraphics[width=209pt]{img-1.eps}
+ \includegraphics[width=209pt]{img2.jpg}
\end{table}
\begin{table}
\centering
- \caption{3 clusters X 67 noeuds}
+ \caption{3 clusters X 67 nodes}
\label{tab.cluster.3x67}
- \includegraphics[width=128pt]{img-2.eps}
+% \includegraphics[width=160pt]{img3.jpg}
+ \includegraphics[scale=0.5]{img3.jpg}
\end{table}
\paragraph*{Interpretations and comments}
\section{Conclusion}
-
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-% where an .eps filename suffix will be assumed under latex,
-% and a .pdf suffix will be assumed for pdflatex; or what has been declared
-% via \DeclareGraphicsExtensions.
-%\caption{Simulation Results}
-%\label{fig_sim}
-%\end{figure}
-
-% Note that IEEE typically puts floats only at the top, even when this
-% results in a large percentage of a column being occupied by floats.
-
-
-% An example of a double column floating figure using two subfigures.
-% (The subfig.sty package must be loaded for this to work.)
-% The subfigure \label commands are set within each subfloat command, the
-% \label for the overall figure must come after \caption.
-% \hfil must be used as a separator to get equal spacing.
-% The subfigure.sty package works much the same way, except \subfigure is
-% used instead of \subfloat.
-%
-%\begin{figure*}[!t]
-%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
-%\label{fig_first_case}}
-%\hfil
-%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
-%\label{fig_second_case}}}
-%\caption{Simulation results}
-%\label{fig_sim}
-%\end{figure*}
-%
-% Note that often IEEE papers with subfigures do not employ subfigure
-% captions (using the optional argument to \subfloat), but instead will
-% reference/describe all of them (a), (b), etc., within the main caption.
-
-
-% An example of a floating table. Note that, for IEEE style tables, the
-% \caption command should come BEFORE the table. Table text will default to
-% \footnotesize as IEEE normally uses this smaller font for tables.
-% The \label must come after \caption as always.
-%
-%\begin{table}[!t]
-%% increase table row spacing, adjust to taste
-%\renewcommand{\arraystretch}{1.3}
-% if using array.sty, it might be a good idea to tweak the value of
-% \extrarowheight as needed to properly center the text within the cells
-%\caption{An Example of a Table}
-%\label{table_example}
-%\centering
-%% Some packages, such as MDW tools, offer better commands for making tables
-%% than the plain LaTeX2e tabular which is used here.
-%\begin{tabular}{|c||c|}
-%\hline
-%One & Two\\
-%\hline
-%Three & Four\\
-%\hline
-%\end{tabular}
-%\end{table}
-
-
-% Note that IEEE does not put floats in the very first column - or typically
-% anywhere on the first page for that matter. Also, in-text middle ("here")
-% positioning is not used. Most IEEE journals/conferences use top floats
-% exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places
-% footnotes above bottom floats. This can be corrected via the \fnbelowfloat
-% command of the stfloats package.
-
-
-
-
-
-
-
-% conference papers do not normally have an appendix
-
-
-% use section* for acknowledgement
\section*{Acknowledgment}
The authors would like to thank...
-
-
-
% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
-%\IEEEtriggeratref{8}
-% The "triggered" command can be changed if desired:
-%\IEEEtriggercmd{\enlargethispage{-5in}}
-
-% references section
-
-% can use a bibliography generated by BibTeX as a .bbl file
-% BibTeX documentation can be easily obtained at:
-% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
-% The IEEEtran BibTeX style support page is at:
-% http://www.michaelshell.org/tex/ieeetran/bibtex/
\bibliographystyle{IEEEtran}
-% argument is your BibTeX string definitions and bibliography database(s)
-\bibliography{bib/hpccBib}
-%
-% <OR> manually copy in the resultant .bbl file
-% set second argument of \begin to the number of references
-% (used to reserve space for the reference number labels box)
-%\begin{thebibliography}{1}
-%
-%\bibitem{IEEEhowto:kopka}
-%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
-% 0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
-%
-%\end{thebibliography}
-
+\bibliography{hpccBib}
-
-
-% that's all folks
\end{document}
-
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% fill-column: 80
+%%% ispell-local-dictionary: "american"
+%%% End:
