X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/8af7de742ea7a8327a7d2fb131d94fca23d31c28..e9e06195c9dfa2fc247a414ba425d21a0d781c10:/hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index c457958..e201e81 100644
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -319,38 +319,26 @@
 
 
 \usepackage[T1]{fontenc}
-\usepackage{ucs}
-%\usepackage[utf8x]{inputenc}
-\usepackage{lmodern}
-\usepackage{color}
-%% Jolis entetes %%
-\usepackage[Glenn]{fncychap}
-%\usepackage{amsmath}
+\usepackage[utf8]{inputenc}
+\usepackage{amsfonts,amssymb}
+\usepackage{amsmath}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
 %\usepackage{amsthm}
-%\usepackage{amsfonts}
-%\usepackage{graphicx}
+\usepackage{graphicx}
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-% Definition des marges
-\usepackage{vmargin}
-\setpapersize[portrait]{A4}
-\usepackage[francais]{babel}
-% Extension pour les graphiques EPS
-%\usepackage[dvips]{graphicx}
-\usepackage[pdftex,final]{graphicx}
+\usepackage[american]{babel}
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
-\usepackage{hyperref}
+%\usepackage{hyperref}
 
-\ifCLASSINFOpdf
-   \usepackage[pdftex]{graphicx}
-   \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
-\else
-\fi
+\algnewcommand\algorithmicinput{\textbf{Input:}}
+\algnewcommand\Input{\item[\algorithmicinput]}
 
+\algnewcommand\algorithmicoutput{\textbf{Output:}}
+\algnewcommand\Output{\item[\algorithmicoutput]}
 
 
-% correct bad hyphenation here
-\hyphenation{op-tical net-works semi-conduc-tor}
 
 
 \begin{document}
@@ -363,9 +351,9 @@
 % 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}
+\author{\IEEEauthorblockN{RaphaÃ«l Couturier and Arnaud Giersch and David Laiymani and Charles Emile Ramamonjisoa}
 \IEEEauthorblockA{Femto-ST Institute - DISC Department\\
-Université de Franche-Comté\\
+UniversitÃ© de Franche-ComtÃ©\\
 Belfort\\
 Email: raphael.couturier@univ-fcomte.fr}
 %\and
@@ -417,23 +405,156 @@ The abstract goes here.
 
 \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... ) 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)
+DÃ©crire le modÃ¨le asynchrone. Je m'en charge (DL)
 
 \section{SimGrid}
 
-Décrire SimGrid (Arnaud)
+DÃ©crire SimGrid (Arnaud)
+
+
+
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Simulation of the multisplitting method}
+%DÃ©crire le problÃ¨me (algo) traitÃ© ainsi que le processus d'adaptation Ã  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{Simulation of the multi-splitting method}
 
-Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 
 \section{Experimental results}
-{\raggedright
+
 When the ``real'' application runs in the simulation environment and produces
 the expected results, varying the input parameters and the program arguments
 allows us to compare outputs from the code execution. We have noticed from this
@@ -447,9 +568,7 @@ algorithm but also to get the main objective of the experimentation of the
 simulation in having an execution time in asynchronous less than in synchronous
 mode, in others words, in having a ``speedup'' less than 1 (Speedup = Execution
 time in synchronous mode / Execution time in asynchronous mode).
-}
 
-{\raggedright
 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
@@ -457,36 +576,25 @@ was to launch the application on clustered network. In this last configuration,
 degrading the inter-cluster network performance will "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.
-}
 
-{\raggedright
 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 1 with a matrix size
+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.
-}
 
-{\raggedright
 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
 clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table 2 which shows the speedups less than 1 with
+permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the speedups less than 1 with
 a matrix size from 62 to 100 elements.
-}
 
-{\raggedright
 In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in Table 3.
-}
+configuration but increasing by two hundreds hosts has been recorded in Table~\ref{tab.cluster.3x67}.
 
-{\raggedright
 Note that the program was run with the following parameters:
-}
 
-%{\raggedright
-\textbullet{} \textbf {SMPI parameters:}
-%}
+\paragraph*{SMPI parameters}
 
 \begin{itemize}
 	\item HOSTFILE : Hosts file description.
@@ -496,9 +604,7 @@ lat latency , ... ).
 \end{itemize}
 
 
-%{\raggedright
-\textbullet{} \textbf {Arguments of the program:}
-%}
+\paragraph*{Arguments of the program}
 
 \begin{itemize}
 	\item Description of the cluster architecture;
@@ -509,34 +615,37 @@ lat latency , ... ).
 	\item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
-\textbf{Table 1}
-
-\textit{{\scriptsize 2 clusters X 50 nodes}}
-\includegraphics[width=209pt]{img-1.eps}
-
-\textbf{Table 2}
+\begin{table}
+  \centering
+  \caption{2 clusters X 50 nodes}
+  \label{tab.cluster.2x50}
+  \includegraphics[width=209pt]{img1.jpg}
+\end{table}
+
+\begin{table}
+  \centering
+  \caption{3 clusters X 33 nodes}
+  \label{tab.cluster.3x33}
+  \includegraphics[width=209pt]{img2.jpg}
+\end{table}
+
+\begin{table}
+  \centering
+  \caption{3 clusters X 67 nodes}
+  \label{tab.cluster.3x67}
+%  \includegraphics[width=160pt]{img3.jpg}
+  \includegraphics[scale=0.5]{img3.jpg}
+\end{table}
+
+\paragraph*{Interpretations and comments}
 
-\textit{{\scriptsize 3 clusters X 33 n\oe{}uds}}
-\includegraphics[width=209pt]{img-1.eps}
-\textbf{Table 3}
-
-\textit{{\scriptsize 3 clusters X 67 noeuds}}
-\includegraphics[width=128pt]{img-2.eps}
-
-{\raggedright
-\textbf{Interpretations and comments}
-}
-
-{\raggedright
 After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Tables 1 and 2), some combinations of the
+clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50} and~\ref{tab.cluster.3x33}), some combinations of the
 used parameters affecting the results have given a speedup less than 1, showing
 the effectiveness of the asynchronous performance compared to the synchronous
 mode.
-}
 
-{\raggedright
-In the case of a two clusters configuration, Table 1 shows that with a
+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
 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
@@ -548,23 +657,18 @@ 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 .
-}
 
-{\raggedright
-For the 3 clusters architecture including a total of 100 hosts, Table 2 shows
+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
 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
 necessary to degrade the inter cluster network bandwidth from 5 to 2 Mbit/s.
-}
 
-{\raggedright
 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 3.
-}
+with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.
 
 \section{Conclusion}
 
@@ -687,7 +791,7 @@ The authors would like to thank...
 % http://www.michaelshell.org/tex/ieeetran/bibtex/
 \bibliographystyle{IEEEtran}
 % argument is your BibTeX string definitions and bibliography database(s)
-\bibliography{bib/hpccBib}
+\bibliography{hpccBib}
 %
 % <OR> manually copy in the resultant .bbl file
 % set second argument of \begin to the number of references
@@ -706,4 +810,9 @@ The authors would like to thank...
 % that's all folks
 \end{document}
 
-
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% fill-column: 80
+%%% ispell-local-dictionary: "american"
+%%% End:
