RCE : Correction parrtie expe

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 7fd9704c72ff315cc3e13f9ff24459a5e1905d38..f9434f99f107282e52b57eb6e0187428cd870c73 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -317,7 +317,7 @@ where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \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 ({\it Generalized Minimal RESidual})~\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}.
+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{algorithm}[t]
 
 \begin{figure}[t]
 %\begin{algorithm}[t]


@@ -386,26 +386,20 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of the two-stage methods using SimGrid toolkit}
 \label{sec:04.02}
 
 \subsection{Simulation of the two-stage methods using SimGrid toolkit}
 \label{sec:04.02}
 

-One of our objectives when simulating the  application in Simgrid is, as in real
+One of our objectives when simulating the  application in SimGrid is, as in real

 life, to  get accurate results  (solutions of the  problem) but also to ensure the
 test reproducibility  under the same  conditions.  According to  our experience,
 life, to  get accurate results  (solutions of the  problem) but also to ensure the
 test reproducibility  under the same  conditions.  According to  our experience,

-very  few modifications  are required  to adapt  a MPI  program for  the Simgrid
+very  few modifications  are required  to adapt  a MPI  program for  the SimGrid

 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI

-libraries  and related  header files  (smpi.h).  The  second modification  is to
+libraries  and related  header files  (\verb+smpi.h+).  The  second modification  is to

 suppress all global variables by replacing  them with local variables or using a
 suppress all global variables by replacing  them with local variables or using a

-Simgrid      selector       called      "runtime       automatic      switching"
+SimGrid selector       called      "runtime       automatic      switching"

 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 effects on runtime between the threads running in the same process and generated by
 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 effects on runtime between the threads running in the same process and generated by

-Simgrid  to simulate the  grid environment.
+SimGrid  to simulate the  grid environment.

 
 

-%\RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
-%last modification on the  MPI program pointed out for some  cases, the review of
-%the sequence of  the MPI\_Isend, MPI\_Irecv and  MPI\_Waitall instructions which
-%might cause an infinite loop.
-
-
-\paragraph{Simgrid Simulator parameters}
-\  \\ \noindent  Before running  a Simgrid  benchmark, many  parameters for  the
+\paragraph{Parameters of the simulation in SimGrid}
+\  \\ \noindent  Before running  a SimGrid  benchmark, many  parameters for  the

 computation platform must be defined. For our experiments, we consider platforms
 in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:
 computation platform must be defined. For our experiments, we consider platforms
 in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:


@@ -413,10 +407,10 @@ inter-cluster communications. In the following, these parameters are described:
 \begin{itemize}
        \item hostfile: hosts description file.
        \item platform: file describing the platform architecture: clusters (CPU power,
 \begin{itemize}
        \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{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}).

        \item archi   : grid computational description (number of clusters, number of
        \item archi   : grid computational description (number of clusters, number of

-nodes/processors for each cluster).
+nodes/processors in each cluster).

 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:
 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:


@@ -424,8 +418,8 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
        \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
        \item inner precision $\TOLG$ and outer precision $\TOLM$,
 \begin{itemize}
        \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
        \item inner precision $\TOLG$ and outer precision $\TOLM$,

-       \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
-       \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+       \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+       \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,

        \item matrix off-diagonal value is fixed to $-1.0$,
        \item number of vectors in matrix $S$ (i.e. value of $s$),
        \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
        \item matrix off-diagonal value is fixed to $-1.0$,
        \item number of vectors in matrix $S$ (i.e. value of $s$),
        \item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,


@@ -434,17 +428,18 @@ In addition, the following arguments are given to the programs at runtime:
        \item execution mode: synchronous or asynchronous.
 \end{itemize}
 
        \item execution mode: synchronous or asynchronous.
 \end{itemize}
 

-It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.

 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 

-\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}
 
 \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:
 
 
 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:


@@ -478,9 +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
 
 \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


@@ -489,10 +484,10 @@ 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
 (resp.  inter-clusters backbone 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). \\


@@ -504,8 +499,7 @@ 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


@@ -518,10 +512,10 @@ 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


@@ -532,8 +526,8 @@ 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.
 


@@ -561,12 +555,12 @@ architectures and scaling up the input matrix size}
 \begin{center}
 \begin{tabular}{r c }
  \hline
 \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
+ 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.}}
 \label{tab:01}
 \end{center}
 \AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
 \label{tab:01}
 \end{center}


@@ -577,13 +571,13 @@ architectures and scaling up the input matrix size}
 
 
 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...}

 
 
 \begin{figure} [ht!]
 
 
 \begin{figure} [ht!]


@@ -596,12 +590,11 @@ multisplitting method.
 \end{figure}
 
 
 \end{figure}
 
 

-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
+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
 (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?}
+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?}

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


@@ -609,20 +602,20 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC
 \begin{center}
 \begin{tabular}{r c }
  \hline
 \begin{center}
 \begin{tabular}{r c }
  \hline

- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+ 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}

 \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 (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...}

 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\%.
 
 


@@ -631,7 +624,7 @@ 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}


@@ -644,7 +637,7 @@ 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}


@@ -680,7 +673,7 @@ 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}


@@ -709,7 +702,7 @@ 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}


@@ -741,7 +734,7 @@ 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}
 


@@ -749,7 +742,7 @@ 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}


@@ -810,7 +803,7 @@ 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$^{-2}$\\