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-
-
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
-%\usepackage{amsmath}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
%\usepackage{amsthm}
-%\usepackage{amsfonts}
-%\usepackage{graphicx}
-%\usepackage{xspace}
+\usepackage{graphicx}
\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}
+\usepackage{url}
+\DeclareUrlCommand\email{\urlstyle{same}}
-\begin{document}
-%
-% paper title
-% can use linebreaks \\ within to get better formatting as desired
-\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+\usepackage[autolanguage,np]{numprint}
+\AtBeginDocument{%
+ \renewcommand*\npunitcommand[1]{\text{#1}}
+ \npthousandthpartsep{}}
+\usepackage{xspace}
+\usepackage[textsize=footnotesize]{todonotes}
+\newcommand{\AG}[2][inline]{%
+ \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
-% 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
-}
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
-% make the title area
-\maketitle
+\begin{document}
+
+\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
+
+\author{%
+ \IEEEauthorblockN{%
+ Charles Emile Ramamonjisoa and
+ David Laiymani and
+ Arnaud Giersch and
+ Lilia Ziane Khodja and
+ Raphaël Couturier
+ }
+ \IEEEauthorblockA{%
+ Femto-ST Institute - DISC Department\\
+ Université de Franche-Comté\\
+ Belfort\\
+ Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+ }
+}
+\maketitle
+\AG{Ordre des autheurs pas définitif}
\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,
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-\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.
+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\dots{}) 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 \emph{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 \emph{synchronous} communication mode
+where a new iteration begin only when all nodes communications are
+completed, either \emph{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 \np[\%]{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 (Simulated 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}
-%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
\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}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
network configuration where the synchronous mode will take advantage on the rapid
exchange of information on such high-speed links. Thus, the methodology adopted
was to launch the application on clustered network. In this last configuration,
-degrading the inter-cluster network performance will "penalize" the synchronous
+degrading the inter-cluster network performance will \emph{penalize} the synchronous
mode allowing to get a speedup lower than 1. This action simulates the case of
clusters linked with long distance network like Internet.
As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
-ranging from Nx = Ny = Nz = 62 to 171 elements or from 62$^{3}$ = 238328 to
-171$^{3}$ = 5,211,000 entries.
+ranging from Nx = Ny = Nz = 62 to 171 elements or from $62^{3} = \np{238328}$ to
+$171^{3} = \np{5211000}$ entries.
Then we have changed the network configuration using three clusters containing
respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
\paragraph*{SMPI parameters}
\begin{itemize}
- \item HOSTFILE : Hosts file description.
+ \item HOSTFILE: Hosts file description.
\item PLATFORM: file description of the platform architecture : clusters (CPU power,
-... ) , intra cluster network description, inter cluster network (bandwidth bw ,
-lat latency , ... ).
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+lat latency, \dots{}).
\end{itemize}
\item Description of the cluster architecture;
\item Maximum number of internal and external iterations;
\item Internal and external precisions;
- \item Matrix size NX , NY and NZ;
+ \item Matrix size NX, NY and NZ;
\item Matrix diagonal value = 6.0;
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
\centering
\caption{2 clusters X 50 nodes}
\label{tab.cluster.2x50}
- \includegraphics[width=209pt]{img-1.eps}
+ \AG{Les images manquent dans le dépôt Git. Si ce sont vraiment des tableaux, utiliser un format vectoriel (eps ou pdf), et surtout pas de jpeg!}
+ \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}
+ \AG{Le fichier manque.}
+ \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}
+ \AG{Le fichier manque.}
+% \includegraphics[width=160pt]{img3.jpg}
+ \includegraphics[scale=0.5]{img3.jpg}
\end{table}
\paragraph*{Interpretations and comments}
mode.
In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
-deterioration of inter cluster network set with 5 Mbits/s of bandwidth, a latency
+deterioration of inter cluster network set with \np[Mbits/s]{5} of bandwidth, a latency
in order of a hundredth of a millisecond and a system power of one GFlops, an
-efficiency of about 40\% in asynchronous mode is obtained for a matrix size of 62
-elements . It is noticed that the result remains stable even if we vary the
-external precision from E -05 to E-09. By increasing the problem size up to 100
-elements, it was necessary to increase the CPU power of 50 \% to 1.5 GFlops for a
+efficiency of about \np[\%]{40} in asynchronous mode is obtained for a matrix size of 62
+elements. It is noticed that the result remains stable even if we vary the
+external precision from \np{E-5} to \np{E-9}. By increasing the problem size up to 100
+elements, it was necessary to increase the CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a
convergence of the algorithm with the same order of asynchronous mode efficiency.
Maintaining such a system power but this time, increasing network throughput
-inter cluster up to 50 Mbits /s, the result of efficiency of about 40\% is
-obtained with high external precision of E-11 for a matrix size from 110 to 150
-side elements .
+inter cluster up to \np[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} is
+obtained with high external precision of \np{E-11} for a matrix size from 110 to 150
+side elements.
For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows
that it was difficult to have a combination which gives an efficiency of
-asynchronous below 80 \%. Indeed, for a matrix size of 62 elements, equality
+asynchronous below \np[\%]{80}. Indeed, for a matrix size of 62 elements, equality
between the performance of the two modes (synchronous and asynchronous) is
-achieved with an inter cluster of 10 Mbits/s and a latency of E- 01 ms. To
-challenge an efficiency by 78\% with a matrix size of 100 points, it was
+achieved with an inter cluster of \np[Mbits/s]{10} and a latency of \np{E-1} ms. To
+challenge an efficiency by \np[\%]{78} with a matrix size of 100 points, it was
necessary to degrade the inter cluster network bandwidth from 5 to 2 Mbit/s.
A last attempt was made for a configuration of three clusters but more power
-with 200 nodes in total. The convergence with a speedup of 90 \% was obtained
-with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.
+with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was obtained
+with a bandwidth of \np[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.
\section{Conclusion}
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