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\begin{document}

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

\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

\RC{Ordre des auteurs pas définitif.}
\begin{abstract}
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}

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... 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) []. 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{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...

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... can lead to very different number of
iterations and so to very different execution times.




\section{SimGrid}

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  
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\vdots & \ddots & \vdots\\
A_{L1} & \cdots & A_{LL}
\end{array} \right)
\times 
\left(\begin{array}{c}
X_1 \\
\vdots\\
X_L
\end{array} \right)
=
\left(\begin{array}{c}
B_1 \\
\vdots\\
B_L
\end{array} \right)\] 
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}
\left\{
\begin{array}{l}
A_{ll}X_l = Y_l \mbox{,~such that}\\
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}
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 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}{$x^0$, $k$}
\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}
\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 $6$ 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.}
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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.







\section{Experimental results}

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 $N_x = N_y = N_z =
62 \text{ to } 171$ 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}
  \renewcommand{\arraystretch}{1.3}

  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
    \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{tabular}

  \smallskip

  \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
    \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{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.

\begin{table}[!t]
  \centering
  \caption{$3$ clusters, each with $33$ nodes}
  \label{tab.cluster.3x33}
  \renewcommand{\arraystretch}{1.3}

  \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
    \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{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}.

\begin{table}[!t]
  \centering
  \caption{3 clusters, each with 66 nodes}
  \label{tab.cluster.3x67}
  \renewcommand{\arraystretch}{1.3}

  \begin{tabular}{|>{\bfseries}r|c|}
    \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{tabular}
\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. 

\section*{Acknowledgment}

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{}}


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