X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/c9f1e655cef3e735867e6000202cb1f982f05d58..8e535991311d246854e7597c06cdee25b67c8419:/hpcc.tex?ds=sidebyside diff --git a/hpcc.tex b/hpcc.tex index 5fbeca1..c9606c3 100644 --- a/hpcc.tex +++ b/hpcc.tex @@ -1,577 +1,661 @@ - -%% 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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One solution is to run the program in a virtual environment +simulating a real interconnected computers architecture. The results are +convincing and useful solutions are obtained with far fewer resources +than in a real platform. However, challenges remain for the convergence +and efficiency of a class of algorithms that concern us here, namely +numerical parallel iterative algorithms executed in asynchronous mode, +especially in a large scale level. Actually, such algorithm requires a +balance and a compromise between computation and communication time +during the execution. Two important factors determine the success of the +experimentation: the convergence of the iterative algorithm on a large +scale and the execution time reduction in asynchronous mode. Once again, +from the current work, a simulated environment like SimGrid provides +accurate results which are difficult or even impossible to obtain in a +physical platform by exploiting the flexibility of the simulator on the +computing units clusters and the network structure design. Our +experimental outputs showed a saving of up to \np[\%]{40} for the algorithm +execution time in asynchronous mode compared to the synchronous one with +a residual precision up to \np{E-11}. Such successful results open +perspectives on experimentations for running the algorithm on a +simulated large scale growing environment and with larger problem size. + +% no keywords for IEEE conferences +% Keywords: Algorithm distributed iterative asynchronous simulation SimGrid \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 - - - - -% 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. +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 \emph{numerical iterative algorithms} executed in a distributed environment. As their name +suggests, these algorithms solve a given problem 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~\cite{BT89,Bahi07}. + +Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will +be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new +iteration starts and until the approximate solution is reached. These parallel computations can be performed either in +\emph{synchronous} mode where a new iteration begins only when all nodes communications are completed, +or in \emph{asynchronous} mode where processors can continue independently with few or no synchronization points. For +instance in the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model~\cite{bcvc06:ij}, local +computations do not need to wait for required data. Processors can then perform their iterations with the data present +at that time. Even if the number of iterations required before the convergence is generally greater than for the +synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to +synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details). + +Parallel numerical applications (synchronous or asynchronous) may have different +configuration and deployment requirements. Quantifying their resource +allocation policies and application scheduling algorithms in grid computing +environments under varying load, CPU power and network speeds is very costly, +very labor intensive and very time +consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC +algorithms is even more problematic since they are very sensible to the +execution environment context. For instance, variations in the network bandwidth +(intra and inter-clusters), in the number and the power of nodes, in the number +of clusters\dots{} can lead to very different number of iterations and so to +very different execution times. Then, it appears that the use of simulation +tools to explore various platform scenarios and to run large numbers of +experiments quickly can be very promising. In this way, 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. + +To our knowledge, there is no existing work on the large-scale simulation of a +real AIAC application. The aim of this paper is twofold. First we give a first +approach of the simulation of AIAC algorithms using a simulation tool (i.e. the +SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of +asynchronous mode algorithms by comparing their performance with the synchronous +mode. More precisely, we had implemented a program for solving large +non-symmetric linear system of equations by numerical method GMRES (Generalized +Minimal Residual) []\AG[]{[]?}. We show, that with minor modifications of the +initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a +real AIAC application on different computing architectures. The simulated +results we obtained are in line with real results exposed in ??\AG[]{??}. +SimGrid 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. 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 is presented with the settings to create various +distributed architectures. The algorithm of the multisplitting method used by GMRES written with MPI primitives and +its adaptation to SimGrid with SMPI (Simulated MPI) is detailed in the next section. At last, the experiments results +carried out will be presented before some concluding remarks and future works. -\section{The asynchronous iteration model} +\section{Motivations and scientific context} + +As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be +classified in three main classes depending on how iterations and communications are managed (for more details readers +can refer to~\cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data +are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and +important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications +(SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously +i.e. without stopping current computations. This technique allows to partially overlap communications by computations +but unfortunately, the overlapping is only partial and important idle times remain. It is clear that, in a grid +computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle +times generated by synchronizations are very penalizing. One way to overcome this problem is to use the +\textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to +wait for required data. Processors can then perform their iterations with the data present at that time. Figure~\ref{fig:aiac} +illustrates this model where the gray blocks represent the computation phases, the white spaces the idle +times and the arrows the communications. With this algorithmic model, the number of iterations required before the +convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC +algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially +in a grid computing context. + +\begin{figure}[!t] + \centering + \includegraphics[width=8cm]{AIAC.pdf} + \caption{The Asynchronous Iterations~-- Asynchronous Communications model} + \label{fig:aiac} +\end{figure} + + +It is very challenging to develop efficient applications for large scale, +heterogeneous and distributed platforms such as computing grids. Researchers and +engineers have to develop techniques for maximizing application performance of +these multi-cluster platforms, by redesigning the applications and/or by using +novel algorithms that can account for the composite and heterogeneous nature of +the platform. Unfortunately, the deployment of such applications on these very +large scale systems is very costly, labor intensive and time consuming. In this +context, it appears that the use of simulation tools to explore various platform +scenarios at will and to run enormous numbers of experiments quickly can be very +promising. Several works\dots{} + +\AG{Several works\dots{} what?\\ + Le paragraphe suivant se trouve déjà dans l'intro ?} +In the context of AIAC algorithms, the use of simulation tools is even more +relevant. Indeed, this class of applications is very sensible to the execution +environment context. For instance, variations in the network bandwidth (intra +and inter-clusters), in the number and the power of nodes, in the number of +clusters\dots{} can lead to very different number of iterations and so to very +different execution times. + -Décrire le modèle asynchrone. Je m'en charge (DL) -\section{SimGrid} -Décrire SimGrid (Arnaud) +\section{SimGrid} -\section{Simulation of the multi-splitting method} +SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation +framework to study the behavior of large-scale distributed systems. As its name +says, it emanates from the grid computing community, but is nowadays used to +study grids, clouds, HPC or peer-to-peer systems. The early versions of SimGrid +date from 1999, but it's still actively developed and distributed as an open +source software. Today, it's one of the major generic tools in the field of +simulation for large-scale distributed systems. + +SimGrid provides several programming interfaces: MSG to simulate Concurrent +Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to +run real applications written in MPI~\cite{MPI}. Apart from the native C +interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming +languages. The SMPI interface supports applications written in C or Fortran, +with little or no modifications. SMPI implements about \np[\%]{80} of the MPI +2.0 standard~\cite{bedaride:hal-00919507}. + +%%% explain simulation +%- simulated processes folded in one real process +%- simulates interactions on the network, fluid model +%- able to skip long-lasting computations +%- traces + visu? + +%%% platforms +%- describe resources and their interconnection, with their properties +%- XML files + +%%% validation + refs + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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 $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping +\begin{equation*} + \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} + B_1 \\ + \vdots\\ + B_L + \end{array} \right) +\end{equation*} +in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$. + +The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system +\begin{equation} + \label{eq:4.1} + \left\{ + \begin{array}{l} + A_{ll}X_l = Y_l \text{, such that}\\ + Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m + \end{array} + \right. +\end{equation} +is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ 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 algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. + +\begin{figure}[!t] + %%% IEEE instructions forbid to use an algorithm environment here, use figure + %%% instead +\begin{algorithmic}[1] +\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector) +\Output $X_l$ (solution sub-vector)\vspace{0.2cm} +\State Load $A_l$, $B_l$ +\State Set the initial guess $x^0$ +\For {$k=0,1,2,\ldots$ until the global convergence} +\State Restart outer iteration with $x^0=x^k$ +\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$} +\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters +\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$ +\EndFor + +\Statex + +\Function {InnerSolver}{$x^0$, $k$} +\State Compute local right-hand side $Y_l$: + \begin{equation*} + Y_l = B_l - \sum\nolimits^L_{\substack{m=1\\ m\neq l}}A_{lm}X_m^0 + \end{equation*} +\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method +\State \Return $X_l^k$ +\EndFunction +\end{algorithmic} +\caption{A multisplitting solver with GMRES method} +\label{algo:01} +\end{figure} + +Algorithm on Figure~\ref{algo:01} shows the main key points of the +multisplitting method to solve a large sparse linear system. This algorithm is +based on an outer-inner iteration method where the parallel synchronous GMRES +method is used to solve the inner iteration. It is executed in parallel by each +cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors +with the subscript $l$ represent the local data for cluster $l$, while +$\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and +$\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with +neighboring clusters. At every outer iteration $k$, asynchronous communications +are performed between processors of the local cluster and those of distant +clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in +Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are +exchanged by message passing using MPI non-blocking communication routines. + +\begin{figure}[!t] +\centering + \includegraphics[width=60mm,keepaspectratio]{clustering} +\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.} +\label{fig:4.1} +\end{figure} + +The global convergence of the asynchronous multisplitting solver is detected +when the clusters of processors have all converged locally. We implemented the +global convergence detection process as follows. On each cluster a master +processor is designated (for example the processor with rank 1) and masters of +all clusters are interconnected by a virtual unidirectional ring network (see +Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around +the virtual ring from a master processor to another until the global convergence +is achieved. So starting from the cluster with rank 1, each master processor $i$ +sets the token to \textit{True} if the local convergence is achieved or to +\text\it{False} otherwise, and sends it to master processor $i+1$. Finally, the +global convergence is detected when the master of cluster 1 receives from the +master of cluster $L$ a token set to \textit{True}. In this case, the master of +cluster 1 broadcasts a stop message to masters of other clusters. In this work, +the local convergence on each cluster $l$ is detected when the following +condition is satisfied +\begin{equation*} + (k\leq \MI) \text{ or } (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon) +\end{equation*} +where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$. + +\LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +We did not encounter major blocking problems when adapting the multisplitting +algorithm previously described to a simulation environment like SimGrid unless +some code debugging. Indeed, apart from the review of the program sequence for +asynchronous exchanges between the six neighbors of each point in a submatrix +within a cluster or between clusters, the algorithm was executed successfully +with SMPI and provided identical outputs as those obtained with direct execution +under MPI. In synchronous mode, the execution of the program raised no +particular issue but in asynchronous mode, the review of the sequence of +MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions and with the addition of +the primitive MPI\_Test was needed to avoid a memory fault due to an infinite +loop resulting from the non-convergence of the algorithm. Note here that the use +of SMPI functions optimizer for memory footprint and CPU usage is not +recommended knowing that one wants to get real results by simulation. As +mentioned, upon this adaptation, the algorithm is executed as in the real life +in the simulated environment after the following minor changes. First, all +declared global variables have been moved to local variables for each +subroutine. In fact, global variables generate side effects arising from the +concurrent access of shared memory used by threads simulating each computing +units in the SimGrid architecture. Second, the alignment of certain types of +variables such as ``long int'' had also to be reviewed. Finally, some +compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed +with the latest version of SimGrid. In total, the initial MPI program running +on the simulation environment SMPI gave after a very simple adaptation the same +results as those obtained in a real environment. We have tested in synchronous +mode with a simulated platform starting from a modest 2 or 3 clusters grid to a +larger configuration like simulating Grid5000 with more than 1500 hosts with +5000 cores~\cite{bolze2006grid}. Once the code debugging and adaptation were +complete, the next section shows our methodology and experimental results. -Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid. \section{Experimental results} -\section{Conclusion} +When the \emph{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 +study that the results depend on the following parameters: +\begin{itemize} +\item At the network level, we found that the most critical values are the + bandwidth (bw) and the network latency (lat). +\item Hosts power (GFlops) can also influence on the results. +\item Finally, when submitting job batches for execution, the arguments values + passed to the program like the maximum number of iterations or the + \emph{external} precision are critical. They allow to ensure not only the + convergence of the 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 (i.e. speed-up less than 1). +\end{itemize} + +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 +\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 50 hosts each, totaling 100 hosts. Various combinations of the above +factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a +matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from +$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} = +\text{\np{5211000}}$ entries. + +% use the same column width for the following three tables +\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}} +\newenvironment{mytable}[1]{% #1: number of columns for data + \renewcommand{\arraystretch}{1.3}% + \begin{tabular}{|>{\bfseries}r% + |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{% + \end{tabular}} + +\begin{table}[!t] + \centering + \caption{2 clusters, each with 50 nodes} + \label{tab.cluster.2x50} + + \begin{mytable}{6} + \hline + bw + & 5 & 5 & 5 & 5 & 5 & 50 \\ + \hline + lat + & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ + \hline + power + & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\ + \hline + size + & 62 & 62 & 62 & 100 & 100 & 110 \\ + \hline + Prec/Eprec + & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\ + \hline + speedup + & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\ + \hline + \end{mytable} + + \smallskip + + \begin{mytable}{6} + \hline + bw + & 50 & 50 & 50 & 50 & 10 & 10 \\ + \hline + lat + & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\ + \hline + power + & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 \\ + \hline + size + & 120 & 130 & 140 & 150 & 171 & 171 \\ + \hline + Prec/Eprec + & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\ + \hline + speedup + & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\ + \hline + \end{mytable} +\end{table} + +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~\ref{tab.cluster.3x33} which shows the +speedups less than 1 with a matrix size from 62 to 100 elements. + +\begin{table}[!t] + \centering + \caption{3 clusters, each with 33 nodes} + \label{tab.cluster.3x33} + + \begin{mytable}{6} + \hline + bw + & 10 & 5 & 4 & 3 & 2 & 6 \\ + \hline + lat + & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ + \hline + power + & 1 & 1 & 1 & 1 & 1 & 1 \\ + \hline + size + & 62 & 100 & 100 & 100 & 100 & 171 \\ + \hline + Prec/Eprec + & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\ + \hline + speedup + & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\ + \hline + \end{mytable} +\end{table} + +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~\ref{tab.cluster.3x67}. + +\begin{table}[!t] + \centering + \caption{3 clusters, each with 66 nodes} + \label{tab.cluster.3x67} + + \begin{mytable}{1} + \hline + bw & 1 \\ + \hline + lat & 0.02 \\ + \hline + power & 1 \\ + \hline + size & 62 \\ + \hline + Prec/Eprec & \np{E-5} \\ + \hline + speedup & 0.9 \\ + \hline + \end{mytable} +\end{table} + +Note that the program was run with the following parameters: + +\paragraph*{SMPI parameters} + +\begin{itemize} + \item HOSTFILE: Hosts file description. + \item PLATFORM: file description of the platform architecture : clusters (CPU power, +\dots{}), intra cluster network description, inter cluster network (bandwidth bw, +lat latency, \dots{}). +\end{itemize} + + +\paragraph*{Arguments of the program} + +\begin{itemize} + \item Description of the cluster architecture; + \item Maximum number of internal and external iterations; + \item Internal and external precisions; + \item Matrix size $N_x$, $N_y$ and $N_z$; + \item Matrix diagonal value: \np{6.0}; + \item Execution Mode: synchronous or asynchronous. +\end{itemize} + +\paragraph*{Interpretations and comments} + +After analyzing the outputs, generally, for the configuration with two or three +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. + +In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows +that with a deterioration of inter cluster network set with \np[Mbit/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 \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 +\np[Mbit/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 \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 +\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. 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 \np[Mbit/s]{2}. + +A last attempt was made for a configuration of three clusters but more powerful +with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was +obtained with a bandwidth of \np[Mbit/s]{1} as shown in +Table~\ref{tab.cluster.3x67}. +\section{Conclusion} +The experimental results on executing a parallel iterative algorithm in +asynchronous mode on an environment simulating a large scale of virtual +computers organized with interconnected clusters have been presented. +Our work has demonstrated that using such a simulation tool allow us to +reach the following three objectives: + +\begin{enumerate} +\item To have a flexible configurable execution platform resolving the +hard exercise to access to very limited but so solicited physical +resources; +\item to ensure the algorithm convergence with a reasonable time and +iteration number ; +\item and finally and more importantly, to find the correct combination +of the cluster and network specifications permitting to save time in +executing the algorithm in asynchronous mode. +\end{enumerate} +Our results have shown that in certain conditions, asynchronous mode is +speeder up to \np[\%]{40} than executing the algorithm in synchronous mode +which is not negligible for solving complex practical problems with more +and more increasing size. + + Several studies have already addressed the performance execution time of +this class of algorithm. The work presented in this paper has +demonstrated an original solution to optimize the use of a simulation +tool to run efficiently an iterative parallel algorithm in asynchronous +mode in a grid architecture. -% 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. However, because -% of issues like this, it may be the safest practice to put all your -% \label just after \caption rather than within \caption{}. -% -% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class -% option should be used if it is desired that the figures are to be -% displayed while in draft mode. -% -%\begin{figure}[!t] -%\centering -%\includegraphics[width=2.5in]{myfigure} -% 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... - - - +This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01). +\todo[inline]{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{IEEEabrv,hpccBib} - -% that's all folks \end{document} - +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% fill-column: 80 +%%% ispell-local-dictionary: "american" +%%% End: + +% LocalWords: Ramamonjisoa Laiymani Arnaud Giersch Ziane Khodja Raphaël Femto +% LocalWords: Université Franche Comté IUT Montbéliard Maréchal Juin Inria Sud +% LocalWords: Ouest Vieille Talence cedex scalability experimentations HPC MPI +% LocalWords: Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua +% LocalWords: Fortran GFlops priori Mbit de du fcomte multisplitting scalable +% LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib