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\begin{document}

\title{Simulation of Asynchronous Iterative Algorithms Using SimGrid}

\author{%
  \IEEEauthorblockN{%
    Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
    Lilia Ziane Khodja\IEEEauthorrefmark{2},
    David Laiymani\IEEEauthorrefmark{1},
    Arnaud Giersch\IEEEauthorrefmark{1} 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

\begin{abstract}

Synchronous  iterative  algorithms  are  often less  scalable  than  asynchronous
iterative  ones.  Performing  large  scale experiments  with  different kind  of
network parameters is not easy  because with supercomputers such parameters are
fixed. So one  solution consists in using simulations first  in order to analyze
what parameters  could influence or not  the behaviors of an  algorithm. In this
paper, we show  that it is interesting to use SimGrid  to simulate the behaviors
of asynchronous  iterative algorithms. For that,  we compare the  behaviour of a
synchronous  GMRES  algorithm  with  an  asynchronous  multisplitting  one  with
simulations  in  which we  choose  some parameters.   Both  codes  are real  MPI
codes. Simulations allow us to see when the multisplitting algorithm can be more
efficient than the GMRES one to solve a 3D Poisson problem.


% 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{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 no
synchronization points~\cite{bcvc06:ij}. In this case, 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,
asynchronous  iterative 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 applications  based on a (synchronous or  asynchronous) iteration model
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 asynchronous
iterative algorithms  is even more problematic  since they are  very sensible to
the  execution environment  context.  For instance,  variations  in the  network
bandwidth (intra and  inter-clusters), in the number and the  power of nodes, in
the number  of clusters\dots{} can lead  to very different  number of iterations
and so  to very  different execution times.   Then, it  appears that the  use of
simulation tools to explore various  platform scenarios and to run large numbers
of  experiments quickly  can  be very  promising.  In  this  way, the  use of  a
simulation  environment  to execute  parallel  iterative  algorithms found  some
interests in reducing the highly cost  of access to computing resources: (1) for
the  applications  development life  cycle  and in  code  debugging  (2) and  in
production  to get  results  in a  reasonable  execution time  with a  simulated
infrastructure not  accessible with physical  resources.  Indeed, the  launch of
distributed  iterative asynchronous  algorithms to  solve a  given problem  on a
large-scale  simulated  environment challenges  to  find optimal  configurations
giving  the best  results  with  a lowest  residual  error and  in  the best  of
execution time.


To our knowledge,  there is no existing work on the  large-scale simulation of a
real asynchronous  iterative application.  {\bf The contribution  of the present
  paper can be  summarised in two main points}.  First we  give a first approach
of the simulation  of asynchronous iterative algorithms using  a simulation tool
(i.e.    the   SimGrid   toolkit~\cite{SimGrid}).    Second,  we   confirm   the
effectiveness  of the  asynchronous  multisplitting algorithm  by comparing  its
performance   with  the   synchronous  GMRES   (Generalized   Minimal  Residual)
\cite{ref1}.  Both  these codes can  be used to  solve large linear  systems. In
this  paper, we  focus  on  a 3D  Poisson  problem.  We  show,  that with  minor
modifications of the initial MPI code,  the SimGrid toolkit allows us to perform
a  test campaign  of  a  real asynchronous  iterative  application on  different
computing architectures.
% The  simulated results  we
%obtained are  in line with real  results exposed in  ??\AG[]{ref?}. 
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.  Parameters  of the
network  platforms  are   the  bandwidth  and  the  latency   of  inter  cluster
network. Parameters on the cluster's architecture are the number of machines and
the  computation power  of a  machine.  Simulations show  that the  asynchronous
multisplitting algorithm  can solve the  3D Poisson problem  approximately twice
faster than GMRES with two distant clusters.



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.  Then, the  multisplitting method is presented, it is
based  on GMRES to  solve each  block obtained  of the  splitting. This  code is
written with MPI  primitives and its adaptation to  SimGrid with SMPI (Simulated
MPI) is  detailed in the next  section. At last, the  simulation 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 synchronous  iterations 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.  It is possible to use asynchronous communications, in this case, the
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 asynchronous iterations 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.  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},
asynchronous  iterative 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 model}
  \label{fig:aiac}
\end{figure}


%% It is very challenging to develop efficient applications for large scale,
%% heterogeneous and distributed platforms such as computing grids. Researchers and
%% engineers have to develop techniques for maximizing application performance of
%% these multi-cluster platforms, by redesigning the applications and/or by using
%% novel algorithms that can account for the composite and heterogeneous nature of
%% the platform. Unfortunately, the deployment of such applications on these very
%% large scale systems is very costly, labor intensive and time consuming. In this
%% context, it appears that the use of simulation tools to explore various platform
%% scenarios at will and to run enormous numbers of experiments quickly can be very
%% promising. Several works\dots{}

%% \AG{Several works\dots{} what?\\
%  Le paragraphe suivant se trouve déjà dans l'intro ?}
In the context of asynchronous algorithms, the number of iterations to reach the
convergence depends on  the delay of messages. With  synchronous iterations, the
number of  iterations is exactly  the same than  in the sequential mode  (if the
parallelization process does  not change the algorithm). So  the difficulty with
asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
with real data. In fact, from an execution to another the order of messages will
change and the  number of iterations to reach the  convergence will also change.
According  to all  the parameters  of the  platform (number  of nodes,  power of
nodes,  inter  and  intra clusrters  bandwith  and  latency,  ....) and  of  the
algorithm  (number   of  splitting  with  the   multisplitting  algorithm),  the
multisplitting code  will obtain the solution  more or less  quickly. Or course,
the GMRES method also depends of the same parameters. As it is difficult to have
access to  many clusters,  grids or supercomputers  with many  different network
parameters,  it  is  interesting  to  be  able  to  simulate  the  behaviors  of
asynchronous iterative algoritms before being able to runs real experiments.






\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.  SMPI is the interface that has been used for the work exposed in
this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
Fortran, with little or no modifications.

Within SimGrid, the execution of a distributed application is simulated on a
single machine.  The application code is really executed, but some operations
like the communications are intercepted, and their running time is computed
according to the characteristics of the simulated execution platform.  The
description of this target platform is given as an input for the execution, by
the mean of an XML file.  It describes the properties of the platform, such as
the computing nodes with their computing power, the interconnection links with
their bandwidth and latency, and the routing strategy.  The simulated running
time of the application is computed according to these properties.

To compute the durations of the operations in the simulated world, and to take
into account resource sharing (e.g. bandwidth sharing between competing
communications), SimGrid uses a fluid model.  This allows to run relatively fast
simulations, while still keeping accurate
results~\cite{bedaride:hal-00919507,tomacs13}.  Moreover, depending on the
simulated application, SimGrid/SMPI allows to skip long lasting computations and
to only take their duration into account.  When the real computations cannot be
skipped, but the results have no importance for the simulation results, there is
also the possibility to share dynamically allocated data structures between
several simulated processes, and thus to reduce the whole memory consumption.
These two techniques can help to run simulations at a very large scale.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors~\cite{o1985multi}. In this case, we apply a row-by-row splitting without overlapping  
\begin{equation*}
  \left(\begin{array}{ccc}
      A_{11} & \cdots & A_{1L} \\
      \vdots & \ddots & \vdots\\
      A_{L1} & \cdots & A_{LL}
    \end{array} \right)
  \times
  \left(\begin{array}{c}
      X_1 \\
      \vdots\\
      X_L
    \end{array} \right)
  =
  \left(\begin{array}{c}
      B_1 \\
      \vdots\\
      B_L
    \end{array} \right)
\end{equation*}
in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$, $A_{\ell
  m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each,
and $\sum_{\ell} n_\ell=\sum_{m} n_m=n$.

The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
\begin{equation}
  \label{eq:4.1}
  \left\{
    \begin{array}{l}
      A_{\ell\ell}X_\ell = Y_\ell \text{, such that}\\
      Y_\ell = B_\ell - \displaystyle\sum_{\substack{m=1\\ m\neq \ell}}^{L}A_{\ell m}X_m
    \end{array}
  \right.
\end{equation}
is solved independently by a cluster and communications are required to update
the right-hand side sub-vector $Y_\ell$, 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 $A_\ell$ (sparse sub-matrix), $B_\ell$ (right-hand side sub-vector)
\Output $X_\ell$ (solution sub-vector)\medskip

\State Load $A_\ell$, $B_\ell$
\State Set the initial guess $x^0$
\For {$k=0,1,2,\ldots$ until the global convergence}
\State Restart outer iteration with $x^0=x^k$
\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
\State\label{algo:01:send} Send shared elements of $X_\ell^{k+1}$ to neighboring clusters
\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq \ell}$
\EndFor

\Statex

\Function {InnerSolver}{$x^0$, $k$}
\State Compute local right-hand side $Y_\ell$:
       \begin{equation*}
         Y_\ell = B_\ell - \sum\nolimits^L_{\substack{m=1\\ m\neq \ell}}A_{\ell m}X_m^0
       \end{equation*}
\State Solving sub-system $A_{\ell\ell}X_\ell^k=Y_\ell$ with the parallel GMRES method
\State \Return $X_\ell^k$
\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 $\ell,m\in\{1,\ldots,L\}$, the matrices and
vectors with the subscript $\ell$ represent the local data for cluster $\ell$,
while $\{A_{\ell m}\}_{m\neq \ell}$ are off-diagonal matrices of sparse matrix
$A$ and $\{X_m\}_{m\neq \ell}$ contain vector elements of solution $x$ shared
with neighboring clusters. At every outer iteration $k$, asynchronous
communications are performed between processors of the local cluster and those
of distant clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
exchanged by message passing using MPI non-blocking communication routines.

\begin{figure}[!t]
\centering
  \includegraphics[width=60mm,keepaspectratio]{clustering}
\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
\label{fig:4.1}
\end{figure}

The global convergence of the asynchronous multisplitting solver is detected
when the clusters of processors have all converged locally. We implemented the
global convergence detection process as follows. On each cluster a master
processor is designated (for example the processor with rank 1) and masters of
all clusters are interconnected by a virtual unidirectional ring network (see
Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around
the virtual ring from a master processor to another until the global convergence
is achieved. So starting from the cluster with rank 1, each master processor $i$
sets the token to \textit{True} if the local convergence is achieved or to
\textit{False} otherwise, and sends it to master processor $i+1$. Finally, the
global convergence is detected when the master of cluster 1 receives from the
master of cluster $L$ a token set to \textit{True}. In this case, the master of
cluster 1 broadcasts a stop message to masters of other clusters. In this work,
the local convergence on each cluster $\ell$ is detected when the following
condition is satisfied
\begin{equation*}
  (k\leq \MI) \text{ or } (\|X_\ell^k - X_\ell^{k+1}\|_{\infty}\leq\epsilon)
\end{equation*}
where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the
tolerance threshold of the error computed between two successive local solution
$X_\ell^k$ and $X_\ell^{k+1}$.



In this paper, we solve the 3D Poisson problem whose the mathematical model is 
\begin{equation}
\left\{
\begin{array}{l}
\nabla^2 u = f \text{~in~} \Omega \\
u =0 \text{~on~} \Gamma =\partial\Omega
\end{array}
\right.
\label{eq:02}
\end{equation}
where $\nabla^2$ is the Laplace operator, $f$ and $u$ are real-valued functions, and $\Omega=[0,1]^3$. The spatial discretization with a finite difference scheme reduces problem~(\ref{eq:02}) to a system of sparse linear equations. Our multisplitting method solves the 3D Poisson problem using a seven point stencil whose the general expression could be written as
\begin{equation}
\begin{array}{l}
u(x-1,y,z) + u(x,y-1,z) + u(x,y,z-1)\\+u(x+1,y,z)+u(x,y+1,z)+u(x,y,z+1) \\ -6u(x,y,z)=h^2f(x,y,z),
%u(x,y,z)= & \frac{1}{6}\times [u(x-1,y,z) + u(x+1,y,z) + \\
 %         & u(x,y-1,z) + u(x,y+1,z) + \\
  %        & u(x,y,z-1) + u(x,y,z+1) - \\ & h^2f(x,y,z)],
\end{array}
\label{eq:03}
\end{equation} 
where $h$ is the distance between two adjacent elements in the spatial discretization scheme and the iteration matrix $A$ of size $N_x\times N_y\times N_z$ of the discretized linear system is sparse, symmetric and positive definite. 

The parallel solving of the 3D Poisson problem with our multisplitting method requires a data partitioning of the problem between clusters and between processors within a cluster. We have chosen the 3D partitioning instead of the row-by-row partitioning in order to reduce the data exchanges at sub-domain boundaries. Figure~\ref{fig:4.2} shows an example of the data partitioning of the 3D Poisson problem between two clusters of processors, where each sub-problem is assigned to a processor. In this context, a processor has at most six neighbors within a cluster or in distant clusters with which it shares data at sub-domain boundaries. 

\begin{figure}[!t]
\centering
  \includegraphics[width=80mm,keepaspectratio]{partition}
\caption{Example of the 3D data partitioning between two clusters of processors.}
\label{fig:4.2}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 processors 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.
%\CER{On voulait en fait montrer la simplicité de l'adaptation de l'algo a SimGrid. Les problèmes rencontrés décrits dans ce paragraphe concerne surtout le mode async}\LZK{OK. J'aurais préféré avoir un peu plus de détails sur l'adaptation de la version async} 
%\CER{Le problème majeur sur l'adaptation MPI vers SMPI pour la partie asynchrone de l'algorithme a été le plantage en SMPI de Waitall après un Isend et Irecv. J'avais proposé un workaround en utilisant un MPI\_wait séparé pour chaque échange a la place d'un waitall unique pour TOUTES les échanges, une instruction qui semble bien fonctionner en MPI. Ce workaround aussi fonctionne bien. Mais après, tu as modifié le programme avec l'ajout d'un MPI\_Test, au niveau de la routine de détection de la convergence et du coup, l'échange global avec waitall a aussi fonctionné.}
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. The scope of all declared 
global variables have been moved to local to subroutine. Indeed, global variables generate side effects arising from the concurrent access of 
shared memory used by threads simulating each computing unit in the SimGrid architecture. 
%Second, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
%\AG{compilation or run-time error?}
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 successfully executed the code in synchronous mode using parallel GMRES algorithm compared with our multisplitting algorithm in asynchronous mode after few modifications. 



\section{Simulation results}

When the \textit{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 and the network latency.
\item Hosts processors 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 precision are critical. They allow us to ensure not only the convergence of the
  algorithm but also to get the main objective in getting an execution time in asynchronous communication less than in
  synchronous mode. The ratio between the execution time of synchronous
  compared to the asynchronous mode ($t_\text{sync} / t_\text{async}$) is defined as the \emph{relative gain}. So,
  our objective running the algorithm in SimGrid is to obtain a relative gain
  greater than 1.
\end{itemize}

A priori, obtaining a relative gain greater 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 a clustered network. In this
configuration, degrading the inter-cluster network performance will penalize the
synchronous mode allowing to get a relative gain greater than 1.  This action
simulates the case of distant clusters linked with long distance network as in grid computing context.


% As a first step, 
The algorithm was run on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The algorithm convergence with a 3D
matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
\text{\np{3375000}}$ entries), is obtained in asynchronous in average 2.5 times faster than in the synchronous mode.
\AG{Expliquer comment lire les tableaux.}
\CER{J'ai reformulé la phrase par la lecture du tableau. Plus de détails seront lus dans la partie Interprétations et commentaires}
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\newenvironment{mytable}[1]{% #1: number of columns for data
  \renewcommand{\arraystretch}{1.3}%
  \begin{tabular}{|>{\bfseries}r%
                  |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
    \end{tabular}}

\begin{table}[!t]
  \centering
  \caption{2 clusters, each with 50 nodes}
  \label{tab.cluster.2x50}

  \begin{mytable}{5}
    \hline
    bandwidth (Mbit/s)
    & 5         & 5         & 5         & 5         & 5         \\
    \hline
    latency (ms)
    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      \\
    \hline
    power (GFlops)
    & 1         & 1         & 1         & 1.5       & 1.5       \\
    \hline
    size $(n^3)$
    & 62        & 62        & 62        & 100       & 100       \\
    \hline
    Precision
    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\
    \hline
    \hline
    Relative gain
    & 2.52      & 2.55      & 2.52      & 2.57      & 2.54      \\
    \hline
  \end{mytable}

  \bigskip

  \begin{mytable}{5}
    \hline
    bandwidth (Mbit/s)
    & 50        & 50        & 50        & 50        & 50 \\ %       & 10        & 10 \\
    \hline
    latency (ms)
    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\ %      & 0.03      & 0.01 \\
    \hline
    Power (GFlops)
    & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
    \hline
    size $(n^3)$
    & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
    \hline
    Precision
    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5}  & \np{E-5} \\
    \hline
    \hline
    Relative gain
    & 2.53      & 2.51     & 2.58     & 2.55     & 2.54   \\ %  & 1.59      & 1.29 \\
    \hline
  \end{mytable}
\end{table}
  
%Then we have changed the network configuration using three clusters containing
%respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
%clusters. In the same way as above, a judicious choice of key parameters has
%permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
%relative gains greater than 1 with a matrix size from 62 to 100 elements.

\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
%\begin{table}[!t]
%  \centering
%  \caption{3 clusters, each with 33 nodes}
%  \label{tab.cluster.3x33}
%
%  \begin{mytable}{6}
%    \hline
%    bandwidth 
%    & 10       & 5        & 4        & 3        & 2        & 6 \\
%    \hline
%    latency
%    & 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
%    \hline
%    Relative gain
%    & 1.003    & 1.01     & 1.08     & 1.19     & 1.28     & 1.01 \\
%    \hline
%  \end{mytable}
%\end{table}

%In a final step, results of an execution attempt to scale up the three clustered
%configuration but increasing by two hundreds hosts has been recorded in
%Table~\ref{tab.cluster.3x67}.

%\begin{table}[!t]
%  \centering
%  \caption{3 clusters, each with 66 nodes}
%  \label{tab.cluster.3x67}
%
%  \begin{mytable}{1}
%    \hline
%    bandwidth  & 1 \\
%    \hline
%    latency    & 0.02 \\
%    \hline
%    power      & 1 \\
%    \hline
%    size       & 62 \\
%    \hline
%    Prec/Eprec & \np{E-5} \\
%    \hline
%    \hline
%    Relative gain    & 1.11 \\
%    \hline
%  \end{mytable}
%\end{table}

Note that the program was run with the following parameters:

\paragraph*{SMPI parameters}

\begin{itemize}
\item HOSTFILE: Text file containing the list of the processors units name. Here 100 hosts;
\item PLATFORM: XML file description of the platform architecture : two clusters (cluster1 and cluster2) with the following characteristics :
  \begin{itemize}
  \item Processor unit power: \np[GFlops]{1.5};
  \item Intracluster network bandwidth: \np[Gbit/s]{1.25} and latency:
    \np[$\mu$s]{0.05};
  \item Intercluster network bandwidth: \np[Mbit/s]{5} and latency:
    \np[$\mu$s]{5};
  \end{itemize}
\end{itemize}


\paragraph*{Arguments of the program}

\begin{itemize}
\item Description of the cluster architecture matching the format <Number of
  cluster> <Number of hosts in cluster1> <Number of hosts in cluster2>;
\item Maximum number of iterations;
\item Precisions on the residual error;
\item Matrix size $N_x$, $N_y$ and $N_z$;
\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
\item Matrix off-diagonal value: $-1$;
\item Communication mode: asynchronous.
\end{itemize}

\paragraph*{Interpretations and comments}

After analyzing the outputs, generally, for the two clusters including one hundred hosts configuration (Tables~\ref{tab.cluster.2x50}), some combinations of parameters affecting
the results have given a relative gain more than 2.5, showing the effectiveness of the
asynchronous performance compared to the synchronous mode.

With these settings, Table~\ref{tab.cluster.2x50} shows
that after a deterioration of inter cluster network with a bandwidth of \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
of one GFlops, an efficiency of about \np[\%]{40} is
obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
increasing the matrix size up to 100 elements, it was necessary to increase the
CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency.  Maintaining such processor power but increasing network throughput inter cluster up to
\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 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 a relative gain of asynchronous mode more than 1.2. 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 greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
%inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
  Quelle est la perte de perfs en faisant ça ?}

%A last attempt was made for a configuration of three clusters but more powerful
%with 200 nodes in total. The convergence with a relative gain around 1.1 was
%obtained with a bandwidth of \np[Mbit/s]{1} as shown in
%Table~\ref{tab.cluster.3x67}.

%\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
%\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
%\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
\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. 

\LZK{Perspectives???}

\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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% LocalWords:  Ramamonjisoa Laiymani Arnaud Giersch Ziane Khodja Raphaël Femto
% LocalWords:  Université Franche Comté IUT Montbéliard Maréchal Juin Inria Sud
% LocalWords:  Ouest Vieille Talence cedex scalability experimentations HPC MPI
% LocalWords:  Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
% LocalWords:  Fortran GFlops priori Mbit de du fcomte multisplitting scalable
% LocalWords:  SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib
% LocalWords:  intra durations nonsingular Waitall discretization discretized
% LocalWords:  InnerSolver Isend Irecv