X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/154021e5b0685f238d989478d56a2a6c2d55dc84..d19f827e7641db37b6e37ee51db8e11b4ab92407:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
index efbda8a..cdcce00 100644
--- 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}
+\usepackage{multirow}
+
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
@@ -319,7 +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}.
 
-\begin{figure}[t]
+\begin{figure}[htpb]
 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]
@@ -357,7 +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}).
 
-\begin{figure}[t]
+\begin{figure}[htbp]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
@@ -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}
-	\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 $bw$,
-latency $lat$, \dots{}).
+latency $lat$, \dots{}),
 	\item archi   : grid computational description (number of clusters, number of
 nodes/processors in each cluster).
 \end{itemize}
@@ -433,15 +435,13 @@ It should also be noticed that both solvers have been executed with the SimGrid
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
-\section{Experimental Results}
+\section{Experimental results}
 \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}
-
-
 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
@@ -473,9 +473,9 @@ 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
-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
@@ -483,14 +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. \\
 
-\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
+\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
-(resp.  inter-clusters backbone links). \\
+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
@@ -499,8 +496,7 @@ input data.  \\
 
 \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
@@ -513,10 +509,10 @@ 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}
-\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.
-\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
@@ -527,226 +523,138 @@ 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
- 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.
 
-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
 
-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
-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.
+\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.
 
-The following paragraphs 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}
-\ \\
-% 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 $lat=8\mu$s. In what follows, we will present the test conditions, the output results and our comments. 
 
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{r c }
- \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
- \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 dÃ©fini..}\RC{Nx est dÃ©fini, 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.}}
+\begin{tabular}{ll}
+\hline
+Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\ 
+\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}
 
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
 
+In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of    the    network    between    clusters    is    fixed    to    $N2$    (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 (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).
 
+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16). 
 
+\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 matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
+\end{figure}
 
-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.
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In  Figure~\ref{fig:02} we  present the  execution times  of both  algorithms to
+solve a  3D Poisson problem of  size $150^3$ on two  different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure  that the Krylov two-stage  algorithm is sensitive to  the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an  interesting behavior of  the Krylov  two-stage algorithm. It  is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the  GMRES algorithms.  This  means  that the  multisplitting  methods are  more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des dÃ©cimales par un ``.''}
+\label{fig:02}
+\end{figure}
 
-\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vÃ©rifie...}
-\RC{Les lÃ©gendes ne sont pas explicites...}
+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm. 
 
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impacts on execution times}
+\label{fig:03}
+\end{figure}
 
-\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 sÃ©parateur dÃ©cimal et non la virgule.  Idem dans les autres figures.}}
-  \label{fig:01}
+\subsubsection{Network bandwidth impacts on performances\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impacts on execution time}
+\label{fig:04}
 \end{figure}
 
+\subsubsection{Matrix size impacts on performances\\}
+In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes.  For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem. 
+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. 
 
-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
-(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 trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Problem size impacts on execution times}
+\label{fig:05}
+\end{figure}
 
-\subsubsection{Running on two different inter-clusters network speeds \\}
 
-\begin{table} [ht!]
-\begin{center}
-\begin{tabular}{r c }
- \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
- \end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
-\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 dÃ©finir cela avant...}
-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
-the  network speed  drops down (variation of 12.5\%), the  difference between  the two Multisplitting algorithms execution times can reach more than 25\%.
 
 
 
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
-\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
-\label{fig:02}
-\end{figure}
-%\end{wrapfigure}
 
 
-\subsubsection{Network latency impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\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}
-\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}
-\caption{Network latency impacts on execution time
-\AG{\np{E-6}}}
-\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.   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  prÃ©cÃ©dentes 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 $8.10^{-6}$.
 
-\subsubsection{Network bandwidth impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\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}
-\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
-\label{tab:04}
-\end{table}
 
 
-\begin{figure} [ht!]
-\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.}
-\label{fig:04}
-\end{figure}
 
-The results  of increasing  the network  bandwidth show  the improvement  of the
-performance  for   both  algorithms   by  reducing   the  execution   time  (see
-Figure~\ref{fig:04}). However,  in this  case, the Krylov  multisplitting method
-presents a better  performance in the considered bandwidth interval  with a gain
-of $40\%$ which is only around $24\%$ for the classical GMRES.
 
-\subsubsection{Input matrix size impacts on performance}
-\ \\
-\begin{table} [ht!]
-\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
- \end{tabular}
-\caption{Test conditions: Input matrix size impacts}
-\label{tab:05}
-\end{table}
 
 
-\begin{figure} [ht!]
-\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 drastic increase ($10$ times)  of the number of iterations needed to
-    reach the convergence for the classical GMRES algorithm when the matrix size
-    go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itÃ©rations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
-\item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
-  compared with the Krylov multisplitting method.
-\end{enumerate}
 
-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.
 
-\subsubsection{CPU Power impacts on performance}
+\subsubsection{CPU Power impacts on performance\\}
 
-\begin{table} [ht!]
+
+\begin{table} [htbp]
 \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}
@@ -791,21 +699,23 @@ synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
 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}
-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  ratio  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
- 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$^{-2}$\\
@@ -851,7 +761,7 @@ 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
-    size (N)
+    size ($N^3$)
     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision
