Start of text about SimGrid. More to come.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index 0fdc78763ee92de8176c0d25603a4fd2a3e66496..ab6e020e98836594820e875ab29cc7ab656875d7 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -4,38 +4,99 @@
 \usepackage[utf8]{inputenc}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
 \usepackage[utf8]{inputenc}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}

-\usepackage{algorithm}
+%\usepackage{algorithm}

 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}
 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}

-%\usepackage{xspace}

 \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[american]{babel}
 % 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}}
+
+\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}
+\newcommand{\DL}[2][inline]{%
+  \todo[color=yellow!50,#1]{\sffamily\textbf{DL:} #2}\xspace}
+\newcommand{\LZK}[2][inline]{%
+  \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+  \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
+

 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 

+\newcommand{\MI}{\mathit{MaxIter}}
+

 
 \begin{document}
 
 \title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
 
 
 \begin{document}
 
 \title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
 

-\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}
+\author{%
+  \IEEEauthorblockN{%
+    Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
+    David Laiymani\IEEEauthorrefmark{1},
+    Arnaud Giersch\IEEEauthorrefmark{1},
+    Lilia Ziane Khodja\IEEEauthorrefmark{2} and
+    Raphaël Couturier\IEEEauthorrefmark{1}
+  }
+  \IEEEauthorblockA{\IEEEauthorrefmark{1}%
+    Femto-ST Institute -- DISC Department\\
+    Université de Franche-Comté,
+    IUT de Belfort-Montbéliard\\
+    19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
+    Email: \email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}
+  }
+  \IEEEauthorblockA{\IEEEauthorrefmark{2}%
+    Inria Bordeaux Sud-Ouest\\
+    200 avenue de la Vieille Tour, 33405 Talence cedex, France \\
+    Email: \email{lilia.ziane@inria.fr}
+  }

 }
 
 \maketitle
 
 }
 
 \maketitle
 

+\RC{Ordre des autheurs pas définitif.}

 \begin{abstract}
 \begin{abstract}

-The abstract goes here.
+In recent years, the scalability of large-scale implementation in a 
+distributed environment of algorithms becoming more and more complex has 
+always been hampered by the limits of physical computing resources 
+capacity. 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}
 
 \section{Introduction}
 \end{abstract}
 
 \section{Introduction}


@@ -46,22 +107,22 @@ 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 
 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 
+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 
 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 
+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 
 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 
+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 
 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 
+computations can be performed either in \emph{synchronous} communication mode

 where a new iteration begin only when all nodes communications are 
 where a new iteration begin only when all nodes communications are 

-completed, either "asynchronous" mode where processors can continue 
+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, 
 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, 


@@ -82,8 +143,8 @@ 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 
 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 
+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 
 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 


@@ -91,39 +152,59 @@ 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 
 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 
+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 
 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.
+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. 
 
 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 
+Then, the simulation framework SimGrid will be presented with the

 settings to create various distributed architectures. The algorithm of 
 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 
+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}
 
 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)
+\DL{Décrire le modèle asynchrone. Je m'en charge}

 
 \section{SimGrid}
 
 
 \section{SimGrid}
 

-Décrire SimGrid (Arnaud)
-
-
-
-
-
-
+SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
+framework to sudy 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.
+%- open source, developped since 1999, one of the major solution in the field
+%
+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.
+%- implements most of MPI-2 \cite{ref} standard [CHECK]
+
+%%% 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
+
+\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} (Arnaud)}

 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 %Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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  
+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. In this case, we apply a row-by-row splitting without overlapping  

 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\
 \[
 \left(\begin{array}{ccc}
 A_{11} & \cdots & A_{1L} \\


@@ -138,47 +219,77 @@ X_L
 \end{array} \right)
 =
 \left(\begin{array}{c}
 \end{array} \right)
 =
 \left(\begin{array}{c}

-Y_1 \\
+B_1 \\

 \vdots\\
 \vdots\\

-Y_L
+B_L

 \end{array} \right)\] 
 \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 $\sum_{l} n_l=\sum_{i} n_i=n$.
+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 by $L$ clusters of processors, in such a way each sub-system
+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}
 \left\{
 \begin{array}{l}
 A_{ll}X_l = Y_l \mbox{,~such that}\\
 \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,
+Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m

 \end{array}
 \right.
 \label{eq:4.1}
 \end{equation}
 \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. 
+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{algorithm}
-\caption{A multisplitting solver with inner iteration GMRES method}
+\begin{figure}
+  %%% IEEE instructions forbid to use an algorithm environment here, use figure
+  %%% instead

 \begin{algorithmic}[1]
 \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 $\hat{x}$
+\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 Send shared elements of $X_l^{k+1}$ to neighboring clusters
+\State Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$

 \EndFor
 
 \Statex
 
 \Function {InnerSolver}{$x^0$, $k$}
 \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 Compute local right-hand side $Y_l$: \[Y_l = B_l - \sum\nolimits^L_{\substack{m=1 \\m\neq l}}A_{lm}X_m^0\]
+\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method

 \State \Return $X_l^k$
 \EndFunction
 \end{algorithmic}
 \State \Return $X_l^k$
 \EndFunction
 \end{algorithmic}

+\caption{A multisplitting solver with GMRES method}

 \label{algo:01}
 \label{algo:01}

-\end{algorithm}
+\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 $6$ and $7$ 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}
+\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 {\it True} if the local convergence is achieved or to {\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 {\it 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
+\[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
+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}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 


@@ -190,7 +301,7 @@ is solved independently by a cluster and communication are required to update th
 
 \section{Experimental results}
 
 
 \section{Experimental results}
 

-When the ``real'' application runs in the simulation environment and produces
+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: (1) at the network
 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: (1) at the network


@@ -198,25 +309,26 @@ level, we found that the most critical values are the bandwidth (bw) and the
 network latency (lat). (2) Hosts power (GFlops) can also influence on the
 results. And finally, (3) when submitting job batches for execution, the
 arguments values passed to the program like the maximum number of iterations or
 network latency (lat). (2) Hosts power (GFlops) can also influence on the
 results. And finally, (3) when submitting job batches for execution, the
 arguments values passed to the program like the maximum number of iterations or

-the ``external'' precision are critical to ensure not only the convergence of the
+the \emph{external} precision are critical 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
 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).
+mode, in others words, in having a \emph{speedup} less than 1
+({speedup}${}={}${execution time in synchronous mode}${}/{}${execution time in
+asynchronous mode}).

 
 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,
 
 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
 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 $N_x = N_y = N_z = 62 \text{ to } 171$ 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
 
 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


@@ -232,10 +344,10 @@ Note that the program was run with the following parameters:
 \paragraph*{SMPI parameters}
 
 \begin{itemize}
 \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,
        \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}
 
 
 \end{itemize}
 
 


@@ -245,8 +357,8 @@ lat latency , ... ).
        \item Description of the cluster architecture;
        \item Maximum number of internal and external iterations;
        \item Internal and external precisions;
        \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 diagonal value = 6.0;
+       \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}
 
        \item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 


@@ -254,6 +366,10 @@ lat latency , ... ).
   \centering
   \caption{2 clusters X 50 nodes}
   \label{tab.cluster.2x50}
   \centering
   \caption{2 clusters X 50 nodes}
   \label{tab.cluster.2x50}

+  \AG{Ces tableaux (\ref{tab.cluster.2x50}, \ref{tab.cluster.3x33} et
+    \ref{tab.cluster.3x67}) sont affreux. Utiliser un format vectoriel (eps ou
+    pdf) ou, mieux, les réécrire en \LaTeX{}. Réécrire les légendes proprement
+    également (\texttt{\textbackslash{}times} au lieu de \texttt{X} par ex.)}

   \includegraphics[width=209pt]{img1.jpg}
 \end{table}
 
   \includegraphics[width=209pt]{img1.jpg}
 \end{table}
 


@@ -261,6 +377,7 @@ lat latency , ... ).
   \centering
   \caption{3 clusters X 33 nodes}
   \label{tab.cluster.3x33}
   \centering
   \caption{3 clusters X 33 nodes}
   \label{tab.cluster.3x33}

+  \AG{Refaire le tableau.}

   \includegraphics[width=209pt]{img2.jpg}
 \end{table}
 
   \includegraphics[width=209pt]{img2.jpg}
 \end{table}
 


@@ -268,6 +385,7 @@ lat latency , ... ).
   \centering
   \caption{3 clusters X 67 nodes}
   \label{tab.cluster.3x67}
   \centering
   \caption{3 clusters X 67 nodes}
   \label{tab.cluster.3x67}

+  \AG{Refaire le tableau.}

 %  \includegraphics[width=160pt]{img3.jpg}
   \includegraphics[scale=0.5]{img3.jpg}
 \end{table}
 %  \includegraphics[width=160pt]{img3.jpg}
   \includegraphics[scale=0.5]{img3.jpg}
 \end{table}


@@ -281,36 +399,64 @@ 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
 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
 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
 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
 
 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
 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[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 2 Mbit/s.
 
 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}.
+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[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.

 
 \section{Conclusion}
 
 \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: 
+
+\newcounter{numberedCntD}
+\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 raisonnable 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.
+\setcounter{numberedCntD}{\theenumi}
+\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. 

 
 \section*{Acknowledgment}
 
 
 
 \section*{Acknowledgment}
 
 

-The authors would like to thank...
+The authors would like to thank\dots{}

 
 
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@@ -318,7 +464,7 @@ The authors would like to thank...
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