Suite corrections expés

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index efbda8abecc21ae9c8ba61374eb87746419f44ef..cb95155ce9eb3dd31616beb0ceb1b3882977226d 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -24,6 +24,8 @@
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}

+\usepackage{multirow}
+

 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}


@@ -433,10 +435,10 @@ It should also be noticed that both solvers have been executed with the SimGrid
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 

-\section{Experimental Results}
+\section{Experimental results}

 \label{sec:expe}
 
 \label{sec:expe}
 

-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.

 
 \subsection{The 3D Poisson problem}
 \label{3dpoisson}
 
 \subsection{The 3D Poisson problem}
 \label{3dpoisson}


@@ -473,9 +475,9 @@ have been chosen for the study in this paper. \\
 
 \textbf{Step 2}: Collect the software materials needed for the experimentation.
 In our case, we have two variants algorithms for the resolution of the
 
 \textbf{Step 2}: Collect the software materials needed for the experimentation.
 In our case, we have two variants algorithms for the resolution of the

-3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting
-method. In addition, the Simgrid simulator has been chosen to simulate the
-behaviors of the distributed applications. Simgrid is running in a virtual
+3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+method. In addition, the SimGrid simulator has been chosen to simulate the
+behaviors of the distributed applications. SimGrid is running in a virtual

 machine on a simple laptop. \\
 
 \textbf{Step 3}: Fix the criteria which will be used for the future
 machine on a simple laptop. \\
 
 \textbf{Step 3}: Fix the criteria which will be used for the future


@@ -484,13 +486,15 @@ on the  one hand the algorithm execution mode (synchronous and asynchronous)
 and on the other hand the execution time and the number of iterations to reach the convergence. \\
 
 \textbf{Step 4  }: Set up the  different grid testbed environments  that will be
 and on the other hand the execution time and the number of iterations to reach the convergence. \\
 
 \textbf{Step 4  }: Set up the  different grid testbed environments  that will be

-simulated in the  simulator tool to run the program.  The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+simulated in the  simulator tool to run the program.  The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number

 represents the number  of clusters in the grid and  the second number represents
 represents the number  of clusters in the grid and  the second number represents

-the number  of hosts (processors/cores)  in each  cluster. The network  has been
+the number  of hosts (processors/cores)  in each  cluster. The network has been

 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links

-(resp.  inter-clusters backbone links). \\
+(resp.  inter-clusters backbone links).  \\
+
+\LZK{Il me semble que le bw et lat des deux réseaux varient dans les expés d'une simu à l'autre. On vire la dernière phrase?}

 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially


@@ -499,8 +503,7 @@ input data.  \\
 
 \textbf{Step 6} : Collect and analyze the output results.
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 

-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}

 
 When running a distributed application in a computational grid, many factors may
 have a strong impact on the performance.  First of all, the architecture of the
 
 When running a distributed application in a computational grid, many factors may
 have a strong impact on the performance.  First of all, the architecture of the


@@ -513,10 +516,10 @@ Another important factor  impacting the overall performance  of the application
 is the network configuration. Two main network parameters can modify drastically
 the program output results:
 \begin{enumerate}
 is the network configuration. Two main network parameters can modify drastically
 the program output results:
 \begin{enumerate}

-\item  the network  bandwidth  (bw=bits/s) also  known  as "the  data-carrying
+\item  the network  bandwidth  ($bw$ in bits/s) also  known  as "the  data-carrying

     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.

-\item the  network latency  (lat :  microsecond) defined as  the delay  from the
+\item the  network latency  ($lat$ in microseconds) defined as  the delay  from the

   start time to send  a simple data from a source to a destination.
 \end{enumerate}
 Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
   start time to send  a simple data from a source to a destination.
 \end{enumerate}
 Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster


@@ -527,97 +530,139 @@ and  between distant  clusters.  This parameter is application dependent.
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
  clusters.  In   practice,  these  two   networks  have  different   speeds.
  The intra-network  generally works  like a  high speed  local network  with a
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
  clusters.  In   practice,  these  two   networks  have  different   speeds.
  The intra-network  generally works  like a  high speed  local network  with a

- high bandwith and very low latency. In opposite, the inter-network connects
- clusters sometime via  heterogeneous networks components  throuth internet with
+ high bandwidth and very low latency. In opposite, the inter-network connects
+ clusters sometime via  heterogeneous networks components  through internet with

  a lower speed.  The network  between distant  clusters might  be a  bottleneck
  for  the global performance of the application.
 
  a lower speed.  The network  between distant  clusters might  be a  bottleneck
  for  the global performance of the application.
 

-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+\subsection{Comparison of GMRES and Krylov two-stage algorithms in synchronous mode}

 
 In the scope  of this paper, our  first objective is to analyze  when the Krylov
 
 In the scope  of this paper, our  first objective is to analyze  when the Krylov

-Multisplitting  method   has  better  performance  than   the  classical  GMRES
-method. With a synchronous  iterative method, better performance means a
+two-stage method has  better  performance  than   the  classical  GMRES method. With a synchronous  iterative method, better performance means a

 smaller number of iterations and execution time before reaching the convergence.
 For a systematic study,  the experiments  should figure  out  that, for  various
 grid  parameters values, the simulator will confirm  the targeted outcomes,
 particularly for poor and slow  networks, focusing on the  impact on the
 communication  performance on the chosen class of algorithm.
 smaller number of iterations and execution time before reaching the convergence.
 For a systematic study,  the experiments  should figure  out  that, for  various
 grid  parameters values, the simulator will confirm  the targeted outcomes,
 particularly for poor and slow  networks, focusing on the  impact on the
 communication  performance on the chosen class of algorithm.

+\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}

 
 

-The following paragraphs present the test conditions, the output results
-and our comments.\\
+In what follows, we will present the test conditions, the output results and our comments.\\

 
 

-
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
+%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}

 \ \\
 % environment
 
 \begin{table} [ht!]
 \begin{center}
 \ \\
 % environment
 
 \begin{table} [ht!]
 \begin{center}

-\begin{tabular}{r c }
+\begin{tabular}{ll }

  \hline
  \hline

- Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ Network           & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ \multirow{2}{*}{Matrix size}  & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
+  &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline
+ \end{tabular}
+\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
+\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
+ Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline

  Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
  - &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline
  \end{tabular}
  Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
  - &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline
  \end{tabular}

-\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
+\caption{Test conditions: various grid configurations with the input matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}

 \AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
 \AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}

+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583

 \label{tab:01}
 \end{center}
 \end{table}
 
 \label{tab:01}
 \end{center}
 \end{table}
 

+<<<<<<< HEAD
+In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm.
+%% First,  the results in  Figure~\ref{fig:01}
+%% show for all grid configurations the non-variation of the number of iterations of
+%% classical  GMRES for  a given  input matrix  size; it is not  the case  for the
+%% multisplitting method.
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+\RC{Les légendes ne sont pas explicites...}
+=======

 
 
 
 
 In this  section, we analyze the  performance of algorithms running  on various
 
 
 
 
 In this  section, we analyze the  performance of algorithms running  on various

-grid configurations  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+grid configurations  (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16  and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. First,  the results in  Figure~\ref{fig:01}

 show for all grid configurations the non-variation of the number of iterations of
 classical  GMRES for  a given  input matrix  size; it is not  the case  for the
 multisplitting method.
 
 show for all grid configurations the non-variation of the number of iterations of
 classical  GMRES for  a given  input matrix  size; it is not  the case  for the
 multisplitting method.
 

-\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
-\RC{Les légendes ne sont pas explicites...}
-
+%\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+%\RC{Les légendes ne sont pas explicites...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583

 
 \begin{figure} [ht!]
   \begin{center}
     \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
   \end{center}
 
 \begin{figure} [ht!]
   \begin{center}
     \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
   \end{center}

-  \caption{Various grid configurations with the input matrix size $N_{x}=150$ and $N_{x}=170$\RC{idem}
+  \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$

 \AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}
 \AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}

+\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}

   \label{fig:01}
 \end{figure}
 
   \label{fig:01}
 \end{figure}
 

-
+<<<<<<< HEAD

 The execution  times between  the two algorithms  is significant  with different
 grid architectures, even  with the same number of processors  (for example, 2x16
 The execution  times between  the two algorithms  is significant  with different
 grid architectures, even  with the same number of processors  (for example, 2x16

-and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+and  4x8). We  can  observe  the low  sensitivity  of  the Krylov multisplitting  method

 (compared with the classical GMRES) when scaling up the number of the processors
 in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs
 (compared with the classical GMRES) when scaling up the number of the processors
 in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs

-$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. 
+\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+=======
+
+Secondly, the execution  times between  the two algorithms  is significant  with different
+grid architectures, even  with the same number of processors  (for example, 2 $\times$ 16
+and  4 $\times$ 8). We  can  observ  the sensitivity  of  the Krylov multisplitting  method
+(compared with the classical GMRES) when scaling up the number of the processors
+in the  grid: in  average, the reduction of the execution time for GMRES  (resp. Multisplitting)  algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583

 
 

-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Simulations for two different inter-clusters network speeds \\}

 
 \begin{table} [ht!]
 \begin{center}
 
 \begin{table} [ht!]
 \begin{center}

-\begin{tabular}{r c }
+\begin{tabular}{ll}

  \hline
  \hline

- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture        & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+                          & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ Matrix size              & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
+ \end{tabular}
+\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
+ Inter Networks & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline

  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
  \end{tabular}
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
  \end{tabular}

-\caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with  networks N1 vs N2}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583

 \label{tab:02}
 \end{center}
 \end{table}
 
 \label{tab:02}
 \end{center}
 \end{table}
 

+<<<<<<< HEAD

 These experiments  compare the  behavior of  the algorithms  running first  on a
 These experiments  compare the  behavior of  the algorithms  running first  on a

-speed inter-cluster  network (N1) and  also on  a less performant  network (N2). \RC{Il faut définir cela avant...}
+slow inter-cluster  network (N1) and  also on  a more performant  network (N2). \RC{Il faut définir cela avant...}
+=======
+In this section, the experiments  compare the  behavior of  the algorithms  running on a
+speeder inter-cluster  network (N1) and  also on  a less performant  network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut définir cela avant...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583

 Figure~\ref{fig:02} shows that end users will reduce the execution time
 Figure~\ref{fig:02} shows that end users will reduce the execution time

-for  both  algorithms when using  a  grid  architecture  like  4x16 or  8x8: the reduction is about $2$. The results depict  also that when
+for  both  algorithms when using  a  grid  architecture  like  4 $\times$ 16 or  8 $\times$ 8: the reduction is about $2$. The results depict  also that when

 the  network speed  drops down (variation of 12.5\%), the  difference between  the two Multisplitting algorithms execution times can reach more than 25\%.
 
 
 the  network speed  drops down (variation of 12.5\%), the  difference between  the two Multisplitting algorithms execution times can reach more than 25\%.
 
 


@@ -626,7 +671,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2

 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
 \label{fig:02}
 \end{figure}
 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
 \label{fig:02}
 \end{figure}


@@ -639,7 +684,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline

  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}


@@ -675,7 +720,7 @@ magnitude with a latency of $8.10^{-6}$.
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline

  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}


@@ -704,7 +749,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 4x8\\ %\hline
+ Grid Architecture & 4 $\times$ 8\\ %\hline

  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
  \end{tabular}


@@ -736,7 +781,7 @@ But the interesting results are:
 These  findings may  help a  lot end  users to  setup the  best and  the optimal
 targeted environment for the application deployment when focusing on the problem
 size scale up.  It  should be noticed that the same test has  been done with the
 These  findings may  help a  lot end  users to  setup the  best and  the optimal
 targeted environment for the application deployment when focusing on the problem
 size scale up.  It  should be noticed that the same test has  been done with the

-grid 2x16 leading to the same conclusion.
+grid 2 $\times$ 16 leading to the same conclusion.

 
 \subsubsection{CPU Power impacts on performance}
 
 
 \subsubsection{CPU Power impacts on performance}
 


@@ -744,7 +789,7 @@ grid 2x16 leading to the same conclusion.
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid architecture & 2x16\\ %\hline
+ Grid architecture & 2 $\times$ 16\\ %\hline

  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}


@@ -805,7 +850,7 @@ The test conditions are summarized in the table~\ref{tab:07}: \\
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline

  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\