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[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index cc694247253dc7cb17ec336f4a4509379a16d695..81062dea9d00385f4d765456f1fc4dfb33f5f44b 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -1,3 +1,4 @@
+

 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}
 \documentclass[conference]{IEEEtran}
 
 \usepackage[T1]{fontenc}


@@ -102,51 +103,57 @@ suggests, these algorithms solve a given problem by successive iterations ($X_{n
 $X_{0}$ 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}.
 
 $X_{0}$ 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.
+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
 
 
 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.
+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
 % 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


@@ -173,39 +180,36 @@ 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
 \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
+in  many scientific  domains.   They can  be  classified in  three main  classes

 depending on  how iterations  and communications are  managed (for  more details
 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}
+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}
 
 \begin{figure}[!t]
   \centering
     \includegraphics[width=8cm]{AIAC.pdf}

-  \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
+  \caption{The asynchronous iterations model}

   \label{fig:aiac}
 \end{figure}
 
   \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
 
 %% It is very challenging to develop efficient applications for large scale,
 %% heterogeneous and distributed platforms such as computing grids. Researchers and


@@ -224,7 +228,7 @@ 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
 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
+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
 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


@@ -416,17 +420,17 @@ u =0 \text{~on~} \Gamma =\partial\Omega
 \right.
 \label{eq:02}
 \end{equation}
 \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 
+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{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)),
+\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} 
 \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. 
+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. 
 
 
 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. 
 


@@ -445,14 +449,14 @@ We did not encounter major blocking problems when adapting the multisplitting al
 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.
 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.}
+%\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.
 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, the scope of all declared 
+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. 
 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?}
+%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. 
 
 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. 
 


@@ -470,8 +474,8 @@ study that the results depend on the following parameters:
 \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
 \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,
+  synchronous mode. The ratio between the simulated execution time of synchronous GMRES algorithm
+  compared to the asynchronous multisplitting algorithm ($t_\text{GMRES} / t_\text{Multisplitting}$) 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}
   our objective running the algorithm in SimGrid is to obtain a relative gain
   greater than 1.
 \end{itemize}


@@ -485,14 +489,13 @@ 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.
 
 
 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
+
+Both codes were simulated 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 problem size of the 3D Poisson problem  ranges 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} =
 $\text{62}^\text{3} = \text{\np{238328}}$ 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}
+\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one. 
+%\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
 % 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


@@ -517,7 +520,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       \\
     \hline
     power (GFlops)
     & 1         & 1         & 1         & 1.5       & 1.5       \\
     \hline

-    size
+    size $(n^3)$

     & 62        & 62        & 62        & 100       & 100       \\
     \hline
     Precision
     & 62        & 62        & 62        & 100       & 100       \\
     \hline
     Precision


@@ -542,7 +545,7 @@ $\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
     Power (GFlops)
     & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline
     Power (GFlops)
     & 1.5       & 1.5       & 1.5       & 1.5       & 1.5 \\ %      & 1         & 1.5 \\
     \hline

-    size
+    size $(n^3)$

     & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
     \hline
     Precision
     & 110       & 120       & 130       & 140       & 150  \\ %     & 171       & 171 \\
     \hline
     Precision


@@ -643,8 +646,8 @@ Note that the program was run with the following parameters:
 \item Maximum number of iterations;
 \item Precisions on the residual error;
 \item Matrix size $N_x$, $N_y$ and $N_z$;
 \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: \np{1.0} (See~(\ref{eq:03}));
-\item Matrix off-diagonal value: \np{-1}/\np{6} (See~(\ref{eq:03}));
+\item Matrix diagonal value: $6$ (See~(\ref{eq:03}));
+\item Matrix off-diagonal value: $-1$;

 \item Communication mode: asynchronous.
 \end{itemize}
 
 \item Communication mode: asynchronous.
 \end{itemize}
 


@@ -680,10 +683,10 @@ elements.
 %obtained with a bandwidth of \np[Mbit/s]{1} as shown in
 %Table~\ref{tab.cluster.3x67}.
 
 %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.}
+%\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 
 \section{Conclusion}
 The experimental results on executing a parallel iterative algorithm in 
 asynchronous mode on an environment simulating a large scale of virtual