petites corections

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index d5a458c5ee34b250dcfcf544edcf7ed1eb05f01d..0e69066fc5e1c495315a7ceeaebc33a08d97cfa1 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}}


@@ -405,10 +407,10 @@ in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:
 
 \begin{itemize}
 inter-cluster communications. In the following, these parameters are described:
 
 \begin{itemize}

-       \item hostfile: hosts description file.
+       \item hostfile: hosts description file,

        \item platform: file describing the platform architecture: clusters (CPU power,
 \dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
        \item platform: file describing the platform architecture: clusters (CPU power,
 \dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,

-latency $lat$, \dots{}).
+latency $lat$, \dots{}),

        \item archi   : grid computational description (number of clusters, number of
 nodes/processors in each cluster).
 \end{itemize}
        \item archi   : grid computational description (number of clusters, number of
 nodes/processors in each cluster).
 \end{itemize}


@@ -483,14 +485,17 @@ results comparison and analysis. In the scope of this study, we retain
 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. \\
 
 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
+\textbf{Step 4}: Set up the  different grid testbed environments  that will be

 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$50. The first number
 represents the number  of clusters in the grid and  the second number represents
 the number  of hosts (processors/cores)  in each  cluster. The network has been
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
 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$50. The first number
 represents the number  of clusters in the grid and  the second number represents
 the number  of hosts (processors/cores)  in each  cluster. The network has been
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links

-(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?}
+\RC{il me semble qu'on peut laisser ca}

 
 \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


@@ -531,92 +536,106 @@ and  between distant  clusters.  This parameter is application dependent.
  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.
 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.
+In what follows, we will present the test conditions, the output results and our comments.

 
 

-The following paragraphs present the test conditions, the output results
-and our comments.\\
+%%RAPH : on vire ca, c'est pas clair et pas important
+%For a systematic study,  the experiments  should figure  out  that, for  various
+%grid  parameters values, the simulator will confirm Multisplitting method  better performance compared to classical GMRES, particularly on poor and slow networks.
+%\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}
+%\RCE { Reformule autrement}

 
 
 
 

-\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
 
 \ \\
 % environment
 

+\RC{Je ne comprends plus rien CE : pourquoi dans 5.4.1 il y a 2 network et aussi dans 5.4.2. Quelle est la différence? Dans la figure 3 de la section 5.4.1 pourquoi il n'y a pas N1 et N2?}
+

 \begin{table} [ht!]
 \begin{center}
 \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
- 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
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ \multirow{2}{*}{Network} & Inter (N2): $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+                          & Intra (N1): $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ \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}
  \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?}
-\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
+\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...}
+%\RCE{oui c est precise}

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

-
-
-
-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}
-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  simulations  conducted  on  various  grid
+configurations presented  in Table~\ref{tab:01}. It  should be noticed  that two
+networks are considered: N1 is  the network between clusters (inter-cluster) and
+N2 is the network inside  a cluster (intra-cluster).  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...}
+%\RCE{Corrige}

 
 \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}
-\AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}
+  \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.}
+%\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?}
+  %\RCE {Corrige}
+    \RC{Idéalement dans la légende il faudrait insiquer Pb size=$150^3$ ou $170^3$  car pour l'instant Nx=150 ca n'indique rien concernant Ny et Nz}

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

+

 The execution  times between  the two algorithms  is significant  with different
 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
+grid architectures, even  with the same number of processors  (for example, 2 $\times$ 16
+and  4 $\times  8$). We  can  observe  a better  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 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES. 
+\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?}
+\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant}

 
 

-\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
- - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
+ Grid architecture        & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{Inter 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}
  \end{tabular}

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

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

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

 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 factor is around $2$. 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\%.
 
 


@@ -625,8 +644,9 @@ 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{Various grid configurations with networks N1 vs N2

 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}

+\RCE{Corrige}

 \label{fig:02}
 \end{figure}
 %\end{wrapfigure}
 \label{fig:02}
 \end{figure}
 %\end{wrapfigure}


@@ -638,16 +658,15 @@ 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
- Network & N1 : bw=1Gbs \\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
+ \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline
+                          & $lat$= From 8$\times$10$^{-6}$ to  $6.10^{-5}$ second \\

  Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
 \caption{Test conditions: network latency impacts}
 \label{tab:03}
 \end{table}
 
  Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
 \caption{Test conditions: network latency impacts}
 \label{tab:03}
 \end{table}
 

-
-

 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}


@@ -656,17 +675,13 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \label{fig:03}
 \end{figure}
 
 \label{fig:03}
 \end{figure}
 

-

 According to  the results of  Figure~\ref{fig:03}, a degradation of  the network
 latency from  $8.10^{-6}$ to  $6.10^{-5}$ implies an  absolute time  increase of
 more  than $75\%$  (resp.  $82\%$)  of the  execution  for  the classical  GMRES
 According to  the results of  Figure~\ref{fig:03}, a degradation of  the network
 latency from  $8.10^{-6}$ to  $6.10^{-5}$ implies an  absolute time  increase of
 more  than $75\%$  (resp.  $82\%$)  of the  execution  for  the classical  GMRES

-(resp.  Krylov multisplitting)  algorithm.   In addition,  it  appears that  the
-Krylov multisplitting method tolerates more the network latency variation with a
-less  rate increase  of  the  execution time.\RC{Les  2  précédentes phrases  me
-  semblent en contradiction....}  Consequently, in the worst case ($lat=6.10^{-5
-}$), the  execution time for  GMRES is  almost the double  than the time  of the
-Krylov multisplitting,  even though, the  performance was  on the same  order of
-magnitude with a latency of $8.10^{-6}$.
+(resp.  Krylov multisplitting)  algorithm which means that the GMRES seems tolerate more the network latency variation with a less  rate increase  of  the  execution time. However, the execution time factor between the two algorithms varies from 2.2 to 1.5 times with a network latency decreasing from $8.10^{-6}$ to  $6.10^{-5}$.
+
+\RC{Les  2  précédentes phrases  me  semblent en contradiction....}  
+\RCE{Reformule}

 
 \subsubsection{Network bandwidth impacts on performance}
 \ \\
 
 \subsubsection{Network bandwidth impacts on performance}
 \ \\


@@ -674,11 +689,13 @@ magnitude with a latency of $8.10^{-6}$.
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
+\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline
+                          & $lat$= 5.10$^{-5}$ second \\

  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}
 \caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}
 \caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}

+\RCE{C est le bw}

 \label{tab:04}
 \end{table}
 
 \label{tab:04}
 \end{table}
 


@@ -688,6 +705,7 @@ magnitude with a latency of $8.10^{-6}$.
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impacts on execution time
 \AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impacts on execution time
 \AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}

+\RCE{Corrige}

 \label{fig:04}
 \end{figure}
 
 \label{fig:04}
 \end{figure}
 


@@ -703,9 +721,9 @@ 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
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
- Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
+ Grid Architecture & 4 $\times$ 8\\ %\hline
+ Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
+ Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 40$^{3}$ to 200$^{3}$\\ \hline

  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}
  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}


@@ -725,9 +743,11 @@ In these experiments, the input matrix size  has been set from $N_{x} = N_{y}
 time for  both algorithms increases when  the input matrix size  also increases.
 But the interesting results are:
 \begin{enumerate}
 time for  both algorithms increases when  the input matrix size  also increases.
 But the interesting results are:
 \begin{enumerate}

-  \item the drastic increase ($10$ times)  of the number of iterations needed to
-    reach the convergence for the classical GMRES algorithm when the matrix size
+  \item the important increase ($10$ times)  of the number of iterations needed to
+    reach the convergence for the classical GMRES algorithm particularly, when the matrix size

     go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
     go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire}

+    \RCE{Le nombre d'iterations augmente de 10 fois, cela surtout a partir de N=150}
+    

 \item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
   compared with the Krylov multisplitting method.
 \end{enumerate}
 \item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
   compared with the Krylov multisplitting method.
 \end{enumerate}


@@ -735,7 +755,7 @@ 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}
 


@@ -743,9 +763,10 @@ grid 2x16 leading to the same conclusion.
 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline

- Grid architecture & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
+ Grid architecture & 2 $\times$ 16\\ %\hline
+ Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline
+ Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ 
+ CPU Power & From 3 to 19 GFlops \\ \hline

  \end{tabular}
 \caption{Test conditions: CPU Power impacts}
 \label{tab:06}
  \end{tabular}
 \caption{Test conditions: CPU Power impacts}
 \label{tab:06}


@@ -791,6 +812,7 @@ theoretically reduce  the overall execution  time and can improve  the algorithm
 performance.
 
 \RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
 performance.
 
 \RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}

+\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain}

 In this section, Simgrid simulator tool has been successfully used to show
 the efficiency of  the multisplitting in asynchronous mode and  to find the best
 combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to
 In this section, Simgrid simulator tool has been successfully used to show
 the efficiency of  the multisplitting in asynchronous mode and  to find the best
 combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to


@@ -804,7 +826,7 @@ 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$^{-2}$\\