X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/fadf3e9c2be61f86ddebab4672ca769096168535..65688cc56f78ea56f2e7c39000e5a5ab381ebfd4:/hpcc.tex?ds=inline diff --git a/hpcc.tex b/hpcc.tex index b4d0be3..b847ea5 100644 --- a/hpcc.tex +++ b/hpcc.tex @@ -1,417 +1,213 @@ - -%% bare_conf.tex -%% V1.3 -%% 2007/01/11 -%% by Michael Shell -%% See: -%% http://www.michaelshell.org/ -%% for current contact information. -%% -%% This is a skeleton file demonstrating the use of IEEEtran.cls -%% (requires IEEEtran.cls version 1.7 or later) with an IEEE conference paper. -%% -%% Support sites: -%% http://www.michaelshell.org/tex/ieeetran/ -%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/ -%% and -%% http://www.ieee.org/ - -%%************************************************************************* -%% Legal Notice: -%% This code is offered as-is without any warranty either expressed or -%% implied; without even the implied warranty of MERCHANTABILITY or -%% FITNESS FOR A PARTICULAR PURPOSE! -%% User assumes all risk. -%% In no event shall IEEE or any contributor to this code be liable for -%% any damages or losses, including, but not limited to, incidental, -%% consequential, or any other damages, resulting from the use or misuse -%% of any information contained here. -%% -%% All comments are the opinions of their respective authors and are not -%% necessarily endorsed by the IEEE. -%% -%% This work is distributed under the LaTeX Project Public License (LPPL) -%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used, -%% distributed and modified. 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Do not be tempted to use the -% cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does -% not format its papers in such ways. +\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} +\algnewcommand\algorithmicinput{\textbf{Input:}} +\algnewcommand\Input{\item[\algorithmicinput]} +\algnewcommand\algorithmicoutput{\textbf{Output:}} +\algnewcommand\Output{\item[\algorithmicoutput]} -% *** PDF, URL AND HYPERLINK PACKAGES *** -% -%\usepackage{url} -% url.sty was written by Donald Arseneau. It provides better support for -% handling and breaking URLs. url.sty is already installed on most LaTeX -% systems. The latest version can be obtained at: -% http://www.ctan.org/tex-archive/macros/latex/contrib/misc/ -% Read the url.sty source comments for usage information. Basically, -% \url{my_url_here}. +\begin{document} -% *** Do not adjust lengths that control margins, column widths, etc. *** -% *** Do not use packages that alter fonts (such as pslatex). *** -% There should be no need to do such things with IEEEtran.cls V1.6 and later. -% (Unless specifically asked to do so by the journal or conference you plan -% to submit to, of course. ) +\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: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr} + } +} +\maketitle -\usepackage[T1]{fontenc} -\usepackage[utf8]{inputenc} -%\usepackage{amsmath} -%\usepackage{amsthm} -%\usepackage{amsfonts} -%\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} +\AG{Ne faut-il pas ajouter Lilia en auteur?} +\begin{abstract} +The abstract goes here. +\end{abstract} +\section{Introduction} -\begin{document} -% -% paper title -% can use linebreaks \\ within to get better formatting as desired -\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid} +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} +Décrire le modèle asynchrone. Je m'en charge (DL) -% 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 -} +\section{SimGrid} +Décrire SimGrid (Arnaud) -% 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{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. -% 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 +\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{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{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} @@ -433,15 +229,15 @@ A priori, obtaining a speedup less than 1 would be difficult in a local area 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 @@ -457,10 +253,10 @@ Note that the program was run with the following parameters: \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} @@ -470,7 +266,7 @@ lat latency , ... ). \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} @@ -479,21 +275,25 @@ lat latency , ... ). \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} @@ -505,168 +305,50 @@ the effectiveness of the asynchronous performance compared to the synchronous 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} - -% An example of a floating figure using the graphicx package. -% Note that \label must occur AFTER (or within) \caption. -% For figures, \caption should occur after the \includegraphics. -% Note that IEEEtran v1.7 and later has special internal code that -% is designed to preserve the operation of \label within \caption -% even when the captionsoff option is in effect. 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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... - - - +The authors would like to thank\dots{} % 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} -% -% 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: