modif

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index b9187ddded5fd8a701d00867a98a0f598da751c6..24ab679bc24daa36f551759d838f7ee6726dd8aa 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -321,7 +321,7 @@ A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \end{equation}
 where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
 
 \end{equation}
 where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
 

-\begin{figure}[t]
+\begin{figure}[htpb]

 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]


@@ -359,7 +359,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \end{equation}
 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 
 \end{equation}
 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 

-\begin{figure}[t]
+\begin{figure}[htbp]

 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]


@@ -407,10 +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}


@@ -442,8 +442,6 @@ In this section, experiments for both multisplitting algorithms are reported. Fi
 
 \subsection{The 3D Poisson problem}
 \label{3dpoisson}
 
 \subsection{The 3D Poisson problem}
 \label{3dpoisson}

-
-

 We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
 \begin{equation}
 \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
 We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
 \begin{equation}
 \frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega


@@ -485,17 +483,11 @@ 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$50. The first number
 represents the number  of clusters in the grid and  the second number represents
 simulated in the  simulator tool to run the program.  The following architectures
 have been configured in SimGrid : 2$\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).  \\
-
-\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}
+the number  of hosts (processors/cores)  in each  cluster. \\

 
 \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


@@ -536,60 +528,68 @@ 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 two-stage algorithms in synchronous mode}
-
-In the scope  of this paper, our  first objective is to analyze  when the Krylov
-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.
-In what follows, we will 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}
+\subsection{Comparison between GMRES and two-stage multisplitting algorithms in synchronous mode}
+In the scope of this paper, our first objective is to analyze when the synchronous Krylov 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.

 
 

-
-
-%\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
+Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$. In what follows, we will present the test conditions, the output results and our comments. 

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

-\begin{tabular}{ll }
- \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}
-\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}
+\begin{tabular}{ll}
+\hline
+Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 
+                                        & $N2$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+\multirow{2}{*}{Matrix size}            & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
+                                        & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
+\end{tabular}
+\caption{Parameters for the different simulations}

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

-
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
+\  \\
+% environment

 In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
 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}
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of    the    network    between    clusters    is    fixed    to    $N1$    (see
+Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a
+given matrix size 170$^3$ elements, a  non-variation in the number of iterations
+for the classical GMRES algorithm, which is not the case of the Krylov two-stage
+algorithm. In fact, with multisplitting  algorithms, the number of splitting (in
+our case, it is the number of clusters) influences on the convergence speed. The
+higher the number  of splitting is, the slower the  convergence of the algorithm
+is.

 
 

-\begin{figure} [ht!]
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\begin{figure} [htbp]

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


@@ -603,15 +603,15 @@ the Krylov two-stage algorithm.
 
 
 
 
 
 

-The execution  times between  the two algorithms  is significant  with different
+The execution  times between  both 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  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
 $40\%$ 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. 
 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
 $40\%$ 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}
+%\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{Simulations for two different inter-clusters network speeds \\}
 
 
 \subsubsection{Simulations for two different inter-clusters network speeds \\}
 


@@ -639,12 +639,12 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 
 
 %\begin{wrapfigure}{l}{100mm}
 
 
 %\begin{wrapfigure}{l}{100mm}

-\begin{figure} [ht!]
+\begin{figure} [htbp]

 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Various grid configurations with networks N1 vs N2
-\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
-\RCE{Corrige}
+\caption{Various grid configurations with networks N1 vs N2}
+%\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}


@@ -665,21 +665,22 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \label{tab:03}
 \end{table}
 
 \label{tab:03}
 \end{table}
 

-\begin{figure} [ht!]
+\begin{figure} [htbp]

 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}

-\caption{Network latency impacts on execution time
-\AG{\np{E-6}}}
+\caption{Network latency impacts on execution time}
+%\AG{\np{E-6}}}

 \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
-(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  $6.10^{-5}$.
+In Table~\ref{tab:03}, parameters  for the influence of the  network latency are
+reported.  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. 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}
 \ \\


@@ -692,18 +693,19 @@ more  than $75\%$  (resp.  $82\%$)  of the  execution  for  the classical  GMRES
                           & $lat$= 5.10$^{-5}$ second \\
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}
                           & $lat$= 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}}
-\RCE{C est le bw}
+\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}
 
 

-\begin{figure} [ht!]
+\begin{figure} [htbp]

 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \centering
 \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}
+\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}
 


@@ -721,43 +723,38 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
  \hline
  Grid Architecture & 4 $\times$ 8\\ %\hline
  Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
  \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
+ Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline

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

-\begin{figure} [ht!]
+\begin{figure} [htbp]

 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \caption{Problem size impacts on execution time}
 \label{fig:05}
 \end{figure}
 
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \caption{Problem size impacts on execution time}
 \label{fig:05}
 \end{figure}
 

-In these experiments, the input matrix size  has been set from $N_{x} = N_{y}
-= N_{z} = 40$ to $200$ side elements  that is from $40^{3} = 64.000$ to $200^{3}
-= 8,000,000$  points. Obviously, as  shown in Figure~\ref{fig:05},  the execution
-time for  both algorithms increases when  the input matrix size  also increases.
-But the interesting results are:
-\begin{enumerate}
-  \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}
-    \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}
+In  these  experiments, the  input  matrix  size has  been  set  from $50^3$  to
+$190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both
+algorithms increases when the input matrix size also increases.  For all problem
+sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this
+benchmark, it seems that  the greater the problem size is,  the bigger the ratio
+between both  algorithm execution  times is.  We can also  observ that  for some
+problem   sizes,  the   Krylov   multisplitting  convergence   varies  quite   a
+lot. Consequently the execution times in that cases also varies.
+

 
 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 2 $\times$ 16 leading to the same conclusion.
+grid 4 $\times$ 8 leading to the same conclusion.

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

-\begin{table} [ht!]
+\begin{table} [htbp]

 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline


@@ -809,18 +806,19 @@ synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
 theoretically reduce  the overall execution  time and can improve  the algorithm
 performance.
 
 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}
-\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
-get    the   highest    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
-exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+In this section,  the Simgrid simulator is  used to compare the  behavior of the
+multisplitting in  asynchronous mode  with GMRES  in synchronous  mode.  Several
+benchmarks have  been performed with  various combination of the  grid resources
+(CPU, Network, input  matrix size, \ldots ). The test  conditions are summarized
+in  Table~\ref{tab:07}. In  order to  compare  the execution  times, this  table
+reports the  relative gain between both  algorithms. It is defined  by the ratio
+between  the   execution  time  of   GMRES  and   the  execution  time   of  the
+multisplitting.  The  ration  is  greater  than  one  because  the  asynchronous
+multisplitting version is faster than GMRES.

 
 
 
 

-The test conditions are summarized in the table~\ref{tab:07}: \\

 
 

-\begin{table} [ht!]
+\begin{table} [htbp]

 \centering
 \begin{tabular}{r c }
  \hline
 \centering
 \begin{tabular}{r c }
  \hline


@@ -870,7 +868,7 @@ geographically distant clusters through the internet.
     power (GFlops)
     & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
     \hline
     power (GFlops)
     & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
     \hline

-    size (N)
+    size ($N^3$)

     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision
     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision