[hpcc2014.git] / hpcc.tex

%% bare_conf.tex
%% V1.3
%% 2007/01/11
%% by Michael Shell
%% See:
%% http://www.michaelshell.org/
%% for current contact information.
%%
%% This is a skeleton file demonstrating the use of IEEEtran.cls
%% (requires IEEEtran.cls version 1.7 or later) with an IEEE conference paper.
%%
%% Support sites:
%% http://www.michaelshell.org/tex/ieeetran/
%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/
%% and
%% http://www.ieee.org/

%%*************************************************************************
%% Legal Notice:
%% This code is offered as-is without any warranty either expressed or
%% implied; without even the implied warranty of MERCHANTABILITY or
%% FITNESS FOR A PARTICULAR PURPOSE! 
%% User assumes all risk.
%% In no event shall IEEE or any contributor to this code be liable for
%% any damages or losses, including, but not limited to, incidental,
%% consequential, or any other damages, resulting from the use or misuse
%% of any information contained here.
%%
%% All comments are the opinions of their respective authors and are not
%% necessarily endorsed by the IEEE.
%%
%% This work is distributed under the LaTeX Project Public License (LPPL)
%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
%% distributed and modified. A copy of the LPPL, version 1.3, is included
%% in the base LaTeX documentation of all distributions of LaTeX released
%% 2003/12/01 or later.
%% Retain all contribution notices and credits.
%% ** Modified files should be clearly indicated as such, including  **
%% ** renaming them and changing author support contact information. **
%%
%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
%%                    bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
%%*************************************************************************

% *** Authors should verify (and, if needed, correct) their LaTeX system  ***
% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
% *** with production work. IEEE's font choices can trigger bugs that do  ***
% *** not appear when using other class files.                            ***
% The testflow support page is at:
% http://www.michaelshell.org/tex/testflow/



% Note that the a4paper option is mainly intended so that authors in
% countries using A4 can easily print to A4 and see how their papers will
% look in print - the typesetting of the document will not typically be
% affected with changes in paper size (but the bottom and side margins will).
% Use the testflow package mentioned above to verify correct handling of
% both paper sizes by the user's LaTeX system.
%
% Also note that the "draftcls" or "draftclsnofoot", not "draft", option
% should be used if it is desired that the figures are to be displayed in
% draft mode.
%
\documentclass[conference]{IEEEtran}
% Add the compsoc option for Computer Society conferences.
%
% If IEEEtran.cls has not been installed into the LaTeX system files,
% manually specify the path to it like:
% \documentclass[conference]{../sty/IEEEtran}





% Some very useful LaTeX packages include:
% (uncomment the ones you want to load)


% *** CITATION PACKAGES ***
%
%\usepackage{cite}
% cite.sty was written by Donald Arseneau
% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
% \cite{} output to follow that of IEEE. Loading the cite package will
% result in citation numbers being automatically sorted and properly
% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
% \cite will automatically add leading space, if needed. Use cite.sty's
% noadjust option (cite.sty V3.8 and later) if you want to turn this off.
% cite.sty is already installed on most LaTeX systems. Be sure and use
% version 4.0 (2003-05-27) and later if using hyperref.sty. cite.sty does
% not currently provide for hyperlinked citations.
% The latest version can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/cite/
% The documentation is contained in the cite.sty file itself.






% *** GRAPHICS RELATED PACKAGES ***
%
\ifCLASSINFOpdf
  % \usepackage[pdftex]{graphicx}
  % declare the path(s) where your graphic files are
  % \graphicspath{{../pdf/}{../jpeg/}}
  % and their extensions so you won't have to specify these with
  % every instance of \includegraphics
  % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
\else
  % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
  % will default to the driver specified in the system graphics.cfg if no
  % driver is specified.
  % \usepackage[dvips]{graphicx}
  % declare the path(s) where your graphic files are
  % \graphicspath{{../eps/}}
  % and their extensions so you won't have to specify these with
  % every instance of \includegraphics
  % \DeclareGraphicsExtensions{.eps}
\fi
% graphicx was written by David Carlisle and Sebastian Rahtz. It is
% required if you want graphics, photos, etc. graphicx.sty is already
% installed on most LaTeX systems. The latest version and documentation can
% be obtained at: 
% http://www.ctan.org/tex-archive/macros/latex/required/graphics/
% Another good source of documentation is "Using Imported Graphics in
% LaTeX2e" by Keith Reckdahl which can be found as epslatex.ps or
% epslatex.pdf at: http://www.ctan.org/tex-archive/info/
%
% latex, and pdflatex in dvi mode, support graphics in encapsulated
% postscript (.eps) format. pdflatex in pdf mode supports graphics
% in .pdf, .jpeg, .png and .mps (metapost) formats. Users should ensure
% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
% not a bitmapped formats (.jpeg, .png). IEEE frowns on bitmapped formats
% which can result in "jaggedy"/blurry rendering of lines and letters as
% well as large increases in file sizes.
%
% You can find documentation about the pdfTeX application at:
% http://www.tug.org/applications/pdftex





% *** MATH PACKAGES ***
%
%\usepackage[cmex10]{amsmath}
% A popular package from the American Mathematical Society that provides
% many useful and powerful commands for dealing with mathematics. If using
% it, be sure to load this package with the cmex10 option to ensure that
% only type 1 fonts will utilized at all point sizes. Without this option,
% it is possible that some math symbols, particularly those within
% footnotes, will be rendered in bitmap form which will result in a
% document that can not be IEEE Xplore compliant!
%
% Also, note that the amsmath package sets \interdisplaylinepenalty to 10000
% thus preventing page breaks from occurring within multiline equations. Use:
%\interdisplaylinepenalty=2500
% after loading amsmath to restore such page breaks as IEEEtran.cls normally
% does. amsmath.sty is already installed on most LaTeX systems. The latest
% version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/





% *** SPECIALIZED LIST PACKAGES ***
%
%\usepackage{algorithmic}
% algorithmic.sty was written by Peter Williams and Rogerio Brito.
% This package provides an algorithmic environment fo describing algorithms.
% You can use the algorithmic environment in-text or within a figure
% environment to provide for a floating algorithm. Do NOT use the algorithm
% floating environment provided by algorithm.sty (by the same authors) or
% algorithm2e.sty (by Christophe Fiorio) as IEEE does not use dedicated
% algorithm float types and packages that provide these will not provide
% correct IEEE style captions. The latest version and documentation of
% algorithmic.sty can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithms/
% There is also a support site at:
% http://algorithms.berlios.de/index.html
% Also of interest may be the (relatively newer and more customizable)
% algorithmicx.sty package by Szasz Janos:
% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/




% *** ALIGNMENT PACKAGES ***
%
%\usepackage{array}
% Frank Mittelbach's and David Carlisle's array.sty patches and improves
% the standard LaTeX2e array and tabular environments to provide better
% appearance and additional user controls. As the default LaTeX2e table
% generation code is lacking to the point of almost being broken with
% respect to the quality of the end results, all users are strongly
% advised to use an enhanced (at the very least that provided by array.sty)
% set of table tools. array.sty is already installed on most systems. The
% latest version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/required/tools/


%\usepackage{mdwmath}
%\usepackage{mdwtab}
% Also highly recommended is Mark Wooding's extremely powerful MDW tools,
% especially mdwmath.sty and mdwtab.sty which are used to format equations
% and tables, respectively. The MDWtools set is already installed on most
% LaTeX systems. The lastest version and documentation is available at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/mdwtools/


% IEEEtran contains the IEEEeqnarray family of commands that can be used to
% generate multiline equations as well as matrices, tables, etc., of high
% quality.


%\usepackage{eqparbox}
% Also of notable interest is Scott Pakin's eqparbox package for creating
% (automatically sized) equal width boxes - aka "natural width parboxes".
% Available at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/





% *** SUBFIGURE PACKAGES ***
%\usepackage[tight,footnotesize]{subfigure}
% subfigure.sty was written by Steven Douglas Cochran. This package makes it
% easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
% work, it is a good idea to load it with the tight package option to reduce
% the amount of white space around the subfigures. subfigure.sty is already
% installed on most LaTeX systems. The latest version and documentation can
% be obtained at:
% http://www.ctan.org/tex-archive/obsolete/macros/latex/contrib/subfigure/
% subfigure.sty has been superceeded by subfig.sty.



%\usepackage[caption=false]{caption}
%\usepackage[font=footnotesize]{subfig}
% subfig.sty, also written by Steven Douglas Cochran, is the modern
% replacement for subfigure.sty. However, subfig.sty requires and
% automatically loads Axel Sommerfeldt's caption.sty which will override
% IEEEtran.cls handling of captions and this will result in nonIEEE style
% figure/table captions. To prevent this problem, be sure and preload
% caption.sty with its "caption=false" package option. This is will preserve
% IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later 
% (recommended due to many improvements over 1.2) of subfig.sty supports
% the caption=false option directly:
%\usepackage[caption=false,font=footnotesize]{subfig}
%
% The latest version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
% The latest version and documentation of caption.sty can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/caption/




% *** FLOAT PACKAGES ***
%
%\usepackage{fixltx2e}
% fixltx2e, the successor to the earlier fix2col.sty, was written by
% Frank Mittelbach and David Carlisle. This package corrects a few problems
% in the LaTeX2e kernel, the most notable of which is that in current
% LaTeX2e releases, the ordering of single and double column floats is not
% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
% single column figure to be placed prior to an earlier double column
% figure. The latest version and documentation can be found at:
% http://www.ctan.org/tex-archive/macros/latex/base/



%\usepackage{stfloats}
% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
% the ability to do double column floats at the bottom of the page as well
% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
% LaTeX2e). It also provides a command:
%\fnbelowfloat
% to enable the placement of footnotes below bottom floats (the standard
% LaTeX2e kernel puts them above bottom floats). This is an invasive package
% which rewrites many portions of the LaTeX2e float routines. It may not work
% with other packages that modify the LaTeX2e float routines. The latest
% version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/sttools/
% Documentation is contained in the stfloats.sty comments as well as in the
% presfull.pdf file. Do not use the stfloats baselinefloat ability as IEEE
% does not allow \baselineskip to stretch. Authors submitting work to the
% IEEE should note that IEEE rarely uses double column equations and
% that authors should try to avoid such use. Do not be tempted to use the
% cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does
% not format its papers in such ways.





% *** PDF, URL AND HYPERLINK PACKAGES ***
%
%\usepackage{url}
% url.sty was written by Donald Arseneau. It provides better support for
% handling and breaking URLs. url.sty is already installed on most LaTeX
% systems. The latest version can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/misc/
% Read the url.sty source comments for usage information. Basically,
% \url{my_url_here}.

% *** Do not adjust lengths that control margins, column widths, etc. ***
% *** Do not use packages that alter fonts (such as pslatex).         ***
% There should be no need to do such things with IEEEtran.cls V1.6 and later.
% (Unless specifically asked to do so by the journal or conference you plan
% to submit to, of course. )



\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts,amssymb}
\usepackage{amsmath}
\usepackage{algorithm}
\usepackage{algpseudocode}
%\usepackage{amsthm}
\usepackage{graphicx}
%\usepackage{xspace}
\usepackage[american]{babel}
% Extension pour les graphiques EPS
%\usepackage[dvips]{graphicx}
\usepackage[pdftex,final]{graphicx}
% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}

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

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




\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}


% author names and affiliations
% use a multiple column layout for up to three different
% affiliations
\author{\IEEEauthorblockN{Raphaël Couturier and Arnaud Giersch and David Laiymani and Charles Emile Ramamonjisoa}
\IEEEauthorblockA{Femto-ST Institute - DISC Department\\
Université de Franche-Comté\\
Belfort\\
Email: raphael.couturier@univ-fcomte.fr}
%\and
%\IEEEauthorblockN{Arnaud Giersch}
%\IEEEauthorblockA{Twentieth Century Fox\\
%Springfield, USA\\
%Email: homer@thesimpsons.com}
%\and
%\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
%\IEEEauthorblockA{Starfleet Academy\\
%San Francisco, California 96678-2391\\
%Telephone: (800) 555--1212\\
%Fax: (888) 555--1212
}



% make the title area
\maketitle


\begin{abstract}
%\boldmath
The abstract goes here.
\end{abstract}
% IEEEtran.cls defaults to using nonbold math in the Abstract.
% This preserves the distinction between vectors and scalars. However,
% if the conference you are submitting to favors bold math in the abstract,
% then you can use LaTeX's standard command \boldmath at the very start
% of the abstract to achieve this. Many IEEE journals/conferences frown on
% math in the abstract anyway.

% no keywords




% For peer review papers, you can put extra information on the cover
% page as needed:
% \ifCLASSOPTIONpeerreview
% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
% \fi
%
% For peerreview papers, this IEEEtran command inserts a page break and
% creates the second title. It will be ignored for other modes.
\IEEEpeerreviewmaketitle



\section{Introduction}

Parallel computing and high performance computing (HPC) are becoming 
more and more imperative for solving various problems raised by 
researchers on various scientific disciplines but also by industrial in 
the field. Indeed, the increasing complexity of these requested 
applications combined with a continuous increase of their sizes lead to 
write distributed and parallel algorithms requiring significant hardware 
resources ( grid computing , clusters, broadband network ,etc... ) but 
also a non- negligible CPU execution time. We consider in this paper a 
class of highly efficient parallel algorithms called iterative executed 
in a distributed environment. As their name suggests, these algorithm 
solves a given problem that might be NP- complete complex by successive 
iterations (X$_{n +1 }$= f (X$_{n}$) ) from an initial value X
$_{0}$ to find an approximate value X* of the solution with a very low 
residual error. Several well-known methods demonstrate the convergence 
of these algorithms. Generally, to reduce the complexity and the 
execution time, the problem is divided into several "pieces" that will 
be solved in parallel on multiple processing units. The latter will 
communicate each intermediate results before a new iteration starts 
until the approximate solution is reached. These distributed parallel 
computations can be performed either in "synchronous" communication mode 
where a new iteration begin only when all nodes communications are 
completed, either "asynchronous" mode where processors can continue 
independently without or few synchronization points. Despite the 
effectiveness of iterative approach, a major drawback of the method is 
the requirement of huge resources in terms of computing capacity, 
storage and high speed communication network. Indeed, limited physical 
resources are blocking factors for large-scale deployment of parallel 
algorithms. 

In recent years, the use of a simulation environment to execute parallel 
iterative algorithms found some interests in reducing the highly cost of 
access to computing resources: (1) for the applications development life 
cycle and in code debugging (2) and in production to get results in a 
reasonable execution time with a simulated infrastructure not accessible 
with physical resources. Indeed, the launch of distributed iterative 
asynchronous algorithms to solve a given problem on a large-scale 
simulated environment challenges to find optimal configurations giving 
the best results with a lowest residual error and in the best of 
execution time. According our knowledge, no testing of large-scale 
simulation of the class of algorithm solving to achieve real results has 
been undertaken to date. We had in the scope of this work implemented a 
program for solving large non-symmetric linear system of equations by 
numerical method GMRES (Generalized Minimal Residual ) in the simulation 
environment Simgrid . The simulated platform had allowed us to launch 
the application from a modest computing infrastructure by simulating 
different distributed architectures composed by clusters nodes 
interconnected by variable speed networks. In addition, it has been 
permitted to show the effectiveness of asynchronous mode algorithm by 
comparing its performance with the synchronous mode time. With selected 
parameters on the network platforms (bandwidth, latency of inter cluster 
network) and on the clusters architecture (number, capacity calculation 
power) in the simulated environment , the experimental results have 
demonstrated not only the algorithm convergence within a reasonable time 
compared with the physical environment performance, but also a time 
saving of up to 40 \% in asynchronous mode.

This article is structured as follows: after this introduction, the next 
section will give a brief description of iterative asynchronous model. 
Then, the simulation framework SIMGRID will be presented with the 
settings to create various distributed architectures. The algorithm of 
the multi -splitting method used by GMRES written with MPI primitives 
and its adaptation to Simgrid with SMPI (Simulation MPI ) will be in the 
next section . At last, the experiments results carried out will be 
presented before the conclusion which we will announce the opening of 
our future work after the results.
 
\section{The asynchronous iteration model}

Décrire le modèle asynchrone. Je m'en charge (DL)

\section{SimGrid}

Décrire SimGrid (Arnaud)







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping  
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\vdots & \ddots & \vdots\\
A_{L1} & \cdots & A_{LL}
\end{array} \right)
\times 
\left(\begin{array}{c}
X_1 \\
\vdots\\
X_L
\end{array} \right)
=
\left(\begin{array}{c}
Y_1 \\
\vdots\\
Y_L
\end{array} \right)\] 
in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.

The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ clusters of processors, in such a way each sub-system
\begin{equation}
\left\{
\begin{array}{l}
A_{ll}X_l = Y_l \mbox{,~such that}\\
Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
\end{array}
\right.
\label{eq:4.1}
\end{equation}
is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers. 

\begin{algorithm}
\caption{A multisplitting solver with inner iteration GMRES method}
\begin{algorithmic}[1]
\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
\Output $X_l$ (local solution vector)\vspace{0.2cm}
\State Load $A_l$, $B_l$, $x^0$
\State Initialize the shared vector $\hat{x}=x^0$
\For {$k=1,2,3,\ldots$ until the global convergence}
\State $x^0=\hat{x}$
\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
\EndFor

\Statex

\Function {InnerSolver}{$x^0$, $k$}
\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
\State \Return $X_l^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








\section{Experimental results}

When the ``real'' application runs in the simulation environment and produces
the expected results, varying the input parameters and the program arguments
allows us to compare outputs from the code execution. We have noticed from this
study that the results depend on the following parameters: (1) at the network
level, we found that the most critical values are the bandwidth (bw) and the
network latency (lat). (2) Hosts power (GFlops) can also influence on the
results. And finally, (3) when submitting job batches for execution, the
arguments values passed to the program like the maximum number of iterations or
the ``external'' precision are critical to ensure not only the convergence of the
algorithm but also to get the main objective of the experimentation of the
simulation in having an execution time in asynchronous less than in synchronous
mode, in others words, in having a ``speedup'' less than 1 (Speedup = Execution
time in synchronous mode / Execution time in asynchronous mode).

A priori, obtaining a speedup less than 1 would be difficult in a local area
network configuration where the synchronous mode will take advantage on the rapid
exchange of information on such high-speed links. Thus, the methodology adopted
was to launch the application on clustered network. In this last configuration,
degrading the inter-cluster network performance will "penalize" the synchronous
mode allowing to get a speedup lower than 1. This action simulates the case of
clusters linked with long distance network like Internet.

As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
ranging from Nx = Ny = Nz = 62 to 171 elements or from 62$^{3}$ = 238328 to
171$^{3}$ = 5,211,000 entries.

Then we have changed the network configuration using three clusters containing
respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
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.

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}.

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,
... ) , intra cluster network description, inter cluster network (bandwidth bw ,
lat latency , ... ).
\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 NX , NY and NZ;
        \item Matrix diagonal value = 6.0;
        \item Execution Mode: synchronous or asynchronous.
\end{itemize}

\begin{table}
  \centering
  \caption{2 clusters X 50 nodes}
  \label{tab.cluster.2x50}
  \includegraphics[width=209pt]{img1.jpg}
\end{table}

\begin{table}
  \centering
  \caption{3 clusters X 33 nodes}
  \label{tab.cluster.3x33}
  \includegraphics[width=209pt]{img2.jpg}
\end{table}

\begin{table}
  \centering
  \caption{3 clusters X 67 nodes}
  \label{tab.cluster.3x67}
%  \includegraphics[width=160pt]{img3.jpg}
  \includegraphics[scale=0.5]{img3.jpg}
\end{table}

\paragraph*{Interpretations and comments}

After analyzing the outputs, generally, for the configuration with two or three
clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50} and~\ref{tab.cluster.3x33}), some combinations of the
used parameters affecting the results have given a speedup less than 1, showing
the effectiveness of the asynchronous performance compared to the synchronous
mode.

In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
deterioration of inter cluster network set with 5 Mbits/s of bandwidth, a latency
in order of a hundredth of a millisecond and a system power of one GFlops, an
efficiency of about 40\% in asynchronous mode is obtained for a matrix size of 62
elements . It is noticed that the result remains stable even if we vary the
external precision from E -05 to E-09. By increasing the problem size up to 100
elements, it was necessary to increase the CPU power of 50 \% to 1.5 GFlops for a
convergence of the algorithm with the same order of asynchronous mode efficiency.
Maintaining such a system power but this time, increasing network throughput
inter cluster up to 50 Mbits /s, the result of efficiency of about 40\% is
obtained with high external  precision of E-11 for a matrix size from 110 to 150
side elements .

For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows
that it was difficult to have a combination which gives an efficiency of
asynchronous below 80 \%. Indeed, for a matrix size of 62 elements, equality
between the performance of the two modes (synchronous and asynchronous) is
achieved with an inter cluster of 10 Mbits/s and a latency of E- 01 ms. To
challenge an efficiency by 78\% with a matrix size of 100 points, it was
necessary to degrade the inter cluster network bandwidth from 5 to 2 Mbit/s.

A last attempt was made for a configuration of three clusters but more power
with 200 nodes in total. The convergence with a speedup of 90 \% was obtained
with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.

\section{Conclusion}


% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
% For figures, \caption should occur after the \includegraphics.
% Note that IEEEtran v1.7 and later has special internal code that
% is designed to preserve the operation of \label within \caption
% even when the captionsoff option is in effect. However, because
% of issues like this, it may be the safest practice to put all your
% \label just after \caption rather than within \caption{}.
%
% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
% option should be used if it is desired that the figures are to be
% displayed while in draft mode.
%
%\begin{figure}[!t]
%\centering
%\includegraphics[width=2.5in]{myfigure}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
%\caption{Simulation Results}
%\label{fig_sim}
%\end{figure}

% Note that IEEE typically puts floats only at the top, even when this
% results in a large percentage of a column being occupied by floats.


% An example of a double column floating figure using two subfigures.
% (The subfig.sty package must be loaded for this to work.)
% The subfigure \label commands are set within each subfloat command, the
% \label for the overall figure must come after \caption.
% \hfil must be used as a separator to get equal spacing.
% The subfigure.sty package works much the same way, except \subfigure is
% used instead of \subfloat.
%
%\begin{figure*}[!t]
%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
%\label{fig_second_case}}}
%\caption{Simulation results}
%\label{fig_sim}
%\end{figure*}
%
% Note that often IEEE papers with subfigures do not employ subfigure
% captions (using the optional argument to \subfloat), but instead will
% reference/describe all of them (a), (b), etc., within the main caption.


% An example of a floating table. Note that, for IEEE style tables, the 
% \caption command should come BEFORE the table. Table text will default to
% \footnotesize as IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
%\caption{An Example of a Table}
%\label{table_example}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%\begin{tabular}{|c||c|}
%\hline
%One & Two\\
%\hline
%Three & Four\\
%\hline
%\end{tabular}
%\end{table}


% Note that IEEE does not put floats in the very first column - or typically
% anywhere on the first page for that matter. Also, in-text middle ("here")
% positioning is not used. Most IEEE journals/conferences use top floats
% exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places
% footnotes above bottom floats. This can be corrected via the \fnbelowfloat
% command of the stfloats package.







% conference papers do not normally have an appendix


% use section* for acknowledgement
\section*{Acknowledgment}


The authors would like to thank...





% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
\bibliography{hpccBib}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
%\begin{thebibliography}{1}
%
%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.
%
%\end{thebibliography}




% that's all folks
\end{document}