deplacement de l'explication de poisson dans la partie multisplitting

[hpcc2014.git] / hpcc.tex

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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 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   (synchronous  or  asynchronous)   applications  may   have  different
configuration   and  deployment   requirements.    Quantifying  their   resource
allocation  policies and  application  scheduling algorithms  in grid  computing
environments under  varying load, CPU power  and network speeds  is very costly,
very          labor           intensive          and          very          time
consuming~\cite{Calheiros:2011:CTM:1951445.1951450}.     The   case    of   AIAC
algorithms  is  even  more problematic  since  they  are  very sensible  to  the
execution environment context. For instance, variations in the network bandwidth
(intra and inter-clusters), in the number  and the power of nodes, in the number
of clusters\dots{}  can lead to  very different number  of iterations and  so to
very  different execution times.  Then, it  appears that  the use  of simulation
tools  to  explore  various platform  scenarios  and  to  run large  numbers  of
experiments quickly can be very promising.  In this way, the use of a simulation
environment  to execute parallel  iterative algorithms  found some  interests in
reducing  the  highly  cost  of  access  to computing  resources:  (1)  for  the
applications development life cycle and  in code debugging (2) and in production
to get  results in a reasonable  execution time with  a simulated infrastructure
not  accessible  with physical  resources.  Indeed,  the  launch of  distributed
iterative  asynchronous algorithms  to solve  a given  problem on  a large-scale
simulated environment challenges to  find optimal configurations giving the best
results with a lowest residual error and in the best of execution time.

To our knowledge,  there is no existing work on the  large-scale simulation of a
real  AIAC application.   {\bf  The contribution  of  the present  paper can  be
  summarised  in two  main  points}.  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  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 AIAC
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 \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.  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.
%\LZK{Répétition  par  rapport à l'intro}

\begin{figure}[!t]
  \centering
    \includegraphics[width=8cm]{AIAC.pdf}
  \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
  \label{fig:aiac}
\end{figure}

\RC{Je serais partant de virer AIAC et laisser asynchronous algorithms... à voir}

%% 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 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. The general iteration scheme of our multisplitting method in a 3D domain using a seven point stencil could be written as 
\begin{equation}
\begin{array}{ll}
u^{k+1}(x,y,z)= & u^k(x,y,z) - \frac{1}{6}\times\\
               & (u^k(x-1,y,z) + u^k(x+1,y,z) + \\
               & u^k(x,y-1,z) + u^k(x,y+1,z) + \\
               & u^k(x,y,z-1) + u^k(x,y,z+1)),
\end{array}
\label{eq:03}
\end{equation} 
where 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}
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 unit in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had
also to be reviewed.
\AG{À propos de ces problèmes d'alignement, en dire plus si ça a un intérêt, ou l'enlever.}
 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 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 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 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. The ratio between the execution time of asynchronous
  compared to the synchronous mode is defined as the \emph{relative gain}. So,
  our objective running the algorithm in SimGrid is to obtain a relative gain
  greater than 1.
  \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
    longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
    Ce n'est pas plutôt l'inverse ?}
\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 clustered network. In this last
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 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 provided the results shown in Table~\ref{tab.cluster.2x50} with a
matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
\text{\np{5000211}}$ entries.
\AG{Expliquer comment lire les tableaux.}

% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\newenvironment{mytable}[1]{% #1: number of columns for data
  \renewcommand{\arraystretch}{1.3}%
  \begin{tabular}{|>{\bfseries}r%
                  |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
    \end{tabular}}

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

  \begin{mytable}{6}
    \hline
    bandwidth
    & 5         & 5         & 5         & 5         & 5         & 50 \\
    \hline
    latency
    & 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
    \hline
    Relative gain
    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54     & 2.53 \\
    \hline
  \end{mytable}

  \bigskip

  \begin{mytable}{6}
    \hline
    bandwidth
    & 50        & 50        & 50        & 50        & 10        & 10 \\
    \hline
    latency
    & 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
    \hline
    Relative gain
    & 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.

\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}

~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}
\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, 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 Matrix off-diagonal value: \np{-1.0};
        \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 relative gain more than 2.5, 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 matrix 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 with a relative gain of 1.5\AG[]{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??}

\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
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% LocalWords:  InnerSolver Isend Irecv