X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hpcc2014.git/blobdiff_plain/923e1fe5e990dcc4a47997cc9cfe37cffda3159a..0aa640af7dcb1e33350b4ade113575ce81bb0d81:/hpcc.tex?ds=inline

diff --git a/hpcc.tex b/hpcc.tex
index 02940f0..2c7d65d 100644
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -4,7 +4,7 @@
 \usepackage[utf8]{inputenc}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
-\usepackage{algorithm}
+%\usepackage{algorithm}
 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}
@@ -40,6 +40,10 @@
 
 \newcommand{\MI}{\mathit{MaxIter}}
 
+\usepackage{array}
+\usepackage{color, colortbl}
+\newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}}
+\newcolumntype{Z}[1]{>{\raggedleft}m{#1}}
 
 \begin{document}
 
@@ -101,81 +105,134 @@ simulated large scale growing environment and with larger problem size.
 
 \section{Introduction}
 
-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 \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.
+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 \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
+suggests, these algorithm solves 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{}. 
+
+Parallelization of such algorithms generally involved the division of the problem 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 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. 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{bcvc06:ij} for more details).
+
+Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
+requirements.  Quantifying their performance of resource allocation policies and application scheduling algorithms in
+grid computing environments under varying load, CPU power and network speeds is very costly, labor intensive and time
+consuming \cite{BuRaCa}. 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 bandwith (intra and inter- clusters), in the
+number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to
+very different execution times. In this context, it appears that the use of simulation tools to explore various platform
+scenarios and to run enormous 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 to give a first approach of the simulation of AIAC algorithms using the SimGrid toolkit \cite{SimGrid}. 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). 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. The simulated results we obtained are in line with real results exposed in ?? 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 is 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) 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{bcvc02:ip}). 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 grey 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}
 
-\DL{DÃ©crire le modÃ¨le asynchrone. Je m'en charge}
 
-\section{SimGrid}
+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...
+
+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 bandwith (intra and
+inter-clusters), in the number and the power of nodes, in the number of clusters... can lead to very different number of
+iterations and so to very different execution times.
+
 
-\AG{DÃ©crire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid} (Arnaud)}
 
 
+\section{SimGrid}
+
+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{DÃ©crire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} (Arnaud)}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Simulation of the multisplitting method}
 %DÃ©crire le problÃ¨me (algo) traitÃ© ainsi que le processus d'adaptation Ã  SimGrid.
@@ -212,8 +269,9 @@ Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
 \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{algorithm}
-\caption{A multisplitting solver with GMRES method}
+\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}
@@ -234,12 +292,25 @@ is solved independently by a cluster and communications are required to update t
 \State \Return $X_l^k$
 \EndFunction
 \end{algorithmic}
+\caption{A multisplitting solver with GMRES method}
 \label{algo:01}
-\end{algorithm}
-
-Algorithm~\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 Algorithm~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines. 
+\end{figure}
 
-\begin{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}[!t]
 \centering
   \includegraphics[width=60mm,keepaspectratio]{clustering}
 \caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
@@ -287,18 +358,82 @@ 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 $N_x = N_y = N_z = 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to
-$171^{3} = \np{5211000}$ entries.
+the above factors have providing the results shown in
+Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
+62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
+\np{5211000}$ entries.
+
+\begin{table}[!t]
+  \centering
+  \caption{2 clusters, each with 50 nodes}
+  \label{tab.cluster.2x50}
 
+ \tiny
+ 
+\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|} 
+ \hline 
+ \bf bw & 5 &5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 & 10 & 10\\ 
+ \hline
+ \bf lat & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01\\ 
+ \hline 
+ \bf power & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5\\ \hline    \bf size & 62 & 62 & 62 & 100 & 100 & 110 & 120& 130 & 140 & 150 & 171 & 171\\ \hline
+ \bf Prec/Eprec & 10$^{-5}$ & 10$^{-8}$ & 10$^{-9}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
+ \bf speedup & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778\\ \hline 
+ \end{tabular}
+\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.
+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}
+
+ \tiny
+ 
+\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|} 
+ \hline 
+ \bf bw & 10 &5 & 4 & 3 & 2 & 6\\ \hline
+ \bf lat & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\\ 
+ \hline 
+ \bf power & 1 & 1 & 1 & 1 & 1 & 1\\ \hline    
+ \bf size & 62 & 100 & 100 & 100 & 100 & 171\\ \hline
+ \bf Prec/Eprec & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline 
+ \bf speedup & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99\\ 
+ \hline 
+ \end{tabular}
+\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}.
+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}
+
+ \tiny
+\begin{tabular}{|M{0.55cm}|M{0.25cm}|} 
+ \hline 
+ \bf bw & 1\\ \hline
+ \bf lat & 0.02\\ 
+ \hline 
+ \bf power & 1\\ 
+ \hline    
+ \bf size & 62\\ 
+ \hline
+ \bf Prec/Eprec & 10$^{-5}$\\ 
+ \hline 
+ \bf speedup & 0.9\\ 
+ \hline 
+ \end{tabular}
+\end{table} 
 
 Note that the program was run with the following parameters:
 
@@ -323,66 +458,41 @@ lat latency, \dots{}).
 	\item Execution Mode: synchronous or asynchronous.
 \end{itemize}
 
-\begin{table}
-  \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 rÃ©Ã©crire en \LaTeX{}. RÃ©Ã©crire les lÃ©gendes proprement
-    Ã©galement (\texttt{\textbackslash{}times} au lieu de \texttt{X} par ex.)}
-  \includegraphics[width=209pt]{img1.jpg}
-\end{table}
-
-\begin{table}
-  \centering
-  \caption{3 clusters X 33 nodes}
-  \label{tab.cluster.3x33}
-  \AG{Refaire le tableau.}
-  \includegraphics[width=209pt]{img2.jpg}
-\end{table}
-
-\begin{table}
-  \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}
-
 \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[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 \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[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 \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[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.
+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[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 \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[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 \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[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.
 
 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}.
+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}
 The experimental results on executing a parallel iterative algorithm in 
@@ -425,7 +535,7 @@ The authors would like to thank\dots{}
 % adjust value as needed - may need to be readjusted if
 % the document is modified later
 \bibliographystyle{IEEEtran}
-\bibliography{hpccBib}
+\bibliography{IEEEabrv,hpccBib}
 
 \end{document}
 
