Corrections section 5.4.6

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index f01a0f4c0be843c7d30f735798072ce818b1c9ce..35ab88f5db1f14578b1986ad4f5a986abea301d9 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


@@ -485,16 +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?}
+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


@@ -535,242 +528,115 @@ 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.
-For a systematic study,  the experiments  should figure  out  that, for  various
-grid  parameters values, the simulator will confirm  the targeted outcomes,
-particularly for poor and slow  networks, focusing on the  impact on the
-communication  performance on the chosen class of algorithm.
-\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}

 
 

-In what follows, we will present the test conditions, the output results and our comments.\\
+\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 $lat=8\mu$s. 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
- Network           & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-5}$ \\ %\hline
- \multirow{2}{*}{Matrix size}  & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
-  &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline
- \end{tabular}
-\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
-\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+\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}
 
 \label{tab:01}
 \end{center}
 \end{table}
 

-In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. Figure~\ref{fig:01} shows, for all grid configurations and a given matrix size, a non-variation in the number of iterations for the classical GMRES algorithm, which is not the case of the Krylov two-stage algorithm.
-%% First,  the results in  Figure~\ref{fig:01}
-%% show for all grid configurations the non-variation of the number of iterations of
-%% classical  GMRES for  a given  input matrix  size; it is not  the case  for the
-%% multisplitting method.
-\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
-\RC{Les légendes ne sont pas explicites...}
-
-\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$^3$
-\AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}
-\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
-  \label{fig:01}
-\end{figure}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}

 
 

-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  observe  the low  sensitivity  of  the Krylov multisplitting  method
-(compared with the classical GMRES) when scaling up the number of the processors
-in the  grid: in  average, the GMRES  (resp. Multisplitting)  algorithm performs
-$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. 
-\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
-\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+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).

 
 

-\subsubsection{Simulations for two different inter-clusters network speeds \\}
+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{table} [ht!]
+\begin{figure}[ht]

 \begin{center}
 \begin{center}

-\begin{tabular}{ll}
- \hline
- Grid architecture        & 2$\times$16, 4$\times$8\\ %\hline
- \multirow{2}{*}{Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
-                          & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
- Matrix size              & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
-\label{tab:02}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}

 \end{center}
 \end{center}

-\end{table}
-
-These experiments  compare the  behavior of  the algorithms  running first  on a
-slow inter-cluster  network (N1) and  also on  a more performant  network (N2). \RC{Il faut définir 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\%.
-
-
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
+\end{figure}

 
 

-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\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}
 \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}}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}

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

-%\end{wrapfigure}

 
 

+\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. 

 
 

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

 \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 performances}

 \label{fig:03}
 \end{figure}
 
 \label{fig:03}
 \end{figure}
 

+\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.

 
 

-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 $8.10^{-6}$.
-
-\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!]
+\begin{figure}[ht]

 \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.}
+\caption{Network bandwith impacts on performances}

 \label{fig:04}
 \end{figure}
 
 \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}
-
+\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. 

 
 

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

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

-\caption{Problem size impacts on execution time}
+\caption{Problem size impacts on performances}

 \label{fig:05}
 \end{figure}
 
 \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'itérations 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 performances\\}
+Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.

 
 

-\subsubsection{CPU Power impacts on performance}
-
-\begin{table} [ht!]
-\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
- \end{tabular}
-\caption{Test conditions: CPU Power impacts}
-\label{tab:06}
-\end{table}
-
-\begin{figure} [ht!]
+\begin{figure}[ht]

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

-\caption{CPU Power impacts on execution time}
+\caption{CPU Power impacts on performances}

 \label{fig:06}
 \end{figure}
 \label{fig:06}
 \end{figure}

-
-Using the Simgrid  simulator flexibility, we have tried to  determine the impact
-on the  algorithms performance in  varying the CPU  power of the  clusters nodes
-from $1$ to $19$ GFlops.  The outputs  depicted in Figure~\ref{fig:06}  confirm the
-performance gain,  around $95\%$ for  both of the  two methods, after  adding more
-powerful CPU.

 \ \\
 \ \\

-%\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
-%obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
-%besoin de déployer sur une archi réelle}
-

 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
 with a real platform is most of  the time not possible. Moreover the behavior of
 To conclude these series of experiments, with  SimGrid we have been able to make
 many simulations  with many parameters  variations. Doing all  these experiments
 with a real platform is most of  the time not possible. Moreover the behavior of

-both GMRES and  Krylov multisplitting methods is in accordance  with larger real
-executions on large scale supercomputer~\cite{couturier15}.
+both GMRES and  Krylov two-stage algorithms is in accordance  with larger real
+executions on large scale supercomputers~\cite{couturier15}.

 
 
 \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
 
 
 \subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}


@@ -788,21 +654,23 @@ 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}
-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
 \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$^{-2}$\\


@@ -848,7 +716,7 @@ 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