autres modifs sur les expés

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 827e7e6474ac700a71e9e2251df201401ae15930..a3ede4cda85492d5d1217582010bb994b65bf181 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -114,7 +114,7 @@ their  applications  using  a simulation tool before.
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;

-%performance} 
+%performance}

 \keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle
 \keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle


@@ -131,28 +131,28 @@ are often very important. So, in this context it is difficult to optimize a
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,
 given application for a given  architecture. In this way and in order to reduce
 the access cost to these computing resources it seems very interesting to use a
 simulation environment.  The advantages are numerous: development life cycle,

-code debugging, ability to obtain results quickly,~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.

 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite
 simple. It generally involves the division of the problem into  several
 \emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing
 
 In this paper we focus on a class of highly efficient parallel algorithms called
 \emph{iterative algorithms}. The parallel scheme of iterative methods is quite
 simple. It generally involves the division of the problem into  several
 \emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing

-units.  Each processing unit has to compute an iteration, to send/receive some
+units.  Each processing unit has to compute an iteration to send/receive some

 data dependencies to/from its neighbors and to iterate this process until the
 data dependencies to/from its neighbors and to iterate this process until the

-convergence of the method. Several well-known methods demonstrate the
+convergence of the method. Several well-known studies demonstrate the

 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies
 convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
 task cannot begin a new iteration while it has not received data dependencies

-from its neighbors. We say that the iteration computation follows a synchronous
-scheme. In the asynchronous scheme a task can compute a new iteration without
-having to wait for the data dependencies coming from its neighbors. Both
-communication and computations are asynchronous inducing that there is no more
-idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
-This model presents some advantages and drawbacks that we detail in
-section~\ref{sec:asynchro} but even if the number of iterations required to
-converge is generally  greater  than for the synchronous  case, it appears that
-the asynchronous  iterative scheme  can significantly  reduce  overall execution
-times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
-for more details).
+from its neighbors. We say that the iteration computation follows a
+\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
+iteration without having to wait for the data dependencies coming from its
+neighbors. Both communication and computations are \textit{asynchronous}
+inducing that there is no more idle time, due to synchronizations, between two
+iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
+that we detail in section~\ref{sec:asynchro} but even if the number of
+iterations required to converge is generally  greater  than for the synchronous
+case, it appears that the asynchronous  iterative scheme  can significantly
+reduce  overall execution times by  suppressing idle  times due to
+synchronizations~(see~\cite{bahi07} for more details).

 
 Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
 consuming to find optimal configuration  and deployment requirements for a given
 
 Nevertheless,  in both  cases  (synchronous  or asynchronous)  it  is very  time
 consuming to find optimal configuration  and deployment requirements for a given


@@ -264,7 +264,7 @@ In this paper, we propose two algorithms of two-stage multisplitting methods. Th
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \end{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \end{equation}

-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.

 
 The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
 \begin{equation}
 
 The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
 \begin{equation}


@@ -317,10 +317,12 @@ suppress all global variables by replacing  them with local variables or using a
 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, generated by
 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, generated by

-the Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?}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.
+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}
 
 
 \paragraph{Simgrid Simulator parameters}


@@ -343,9 +345,16 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
        \item maximum number of inner and outer iterations;
        \item inner and outer precisions;
 \begin{itemize}
        \item maximum number of inner and outer iterations;
        \item inner and outer precisions;

-       \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
-       \item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifies, je dis ca de tête}
-       \item execution mode: synchronous or asynchronous.
+       \item maximum number of the gmres's restarts in the Arnorldi process;
+       \item maximum number of iterations qnd the tolerance threshold in classical GMRES;
+       \item tolerance threshold for outer and inner-iterations;
+       \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on x, y, z axis;
+       \item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
+       \item matrix off-diagonal value;
+       \item execution mode: synchronous or asynchronous;
+       \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les  arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
+       \item Size of matrix S;
+       \item Maximum number of iterations and tolerance threshold for CGLS.

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


@@ -356,14 +365,30 @@ It should also be noticed that both solvers have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
 \section{Experimental Results}
 \label{sec:expe}
 

-In this section, experiments for both Multisplitting algorithms are reported. First the problem sued 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{3D Poisson}

 
 

-We use our two-stage algorithms to solve the well-known 3D Poisson problem $\nabla^2\phi=f$, where $\nabla^2$ is the Laplace operator. 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

 \begin{equation}
 \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~}\Omega
+\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

 \label{eq:07}
 \end{equation}
 \label{eq:07}
 \end{equation}

-where the real-valued function $\phi(x,y,z)=0\mbox{~on~}\partial\Omega$ is the solution sought, $f(x,y,z)$ is a known function and the domain $\Omega=[0,1]^3$. 
+such that
+\begin{equation*}
+\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
+\end{equation*}
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
+\begin{equation}
+\begin{array}{ll}
+\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
+\end{array}
+\label{eq:08}
+\end{equation}
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
+
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.

 
 \subsection{Study setup and Simulation Methodology}
 
 
 \subsection{Study setup and Simulation Methodology}
 


@@ -377,7 +402,7 @@ 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
 \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 on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a laptop. \\
+distributed applications. Simgrid is running on the Mesocentre datacenter in the University of  Franche-Comte and also in a virtual machine on a simple laptop. \\

 
 \textbf{Step 3}: Fix the criteria which will be used for the future
 results comparison and analysis. In the scope of this study, we retain
 
 \textbf{Step 3}: Fix the criteria which will be used for the future
 results comparison and analysis. In the scope of this study, we retain


@@ -429,37 +454,35 @@ transit between the clusters and nodes during the code execution.
  In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
  "intra-network" which refers  to the links between nodes within  a cluster and,
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
  In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
  "intra-network" which refers  to the links between nodes within  a cluster and,
  on  the other  hand, the  "inter-network" which  is the  backbone link  between

- clusters.  In   practse;  these  two   networks  have  different   speeds.  The
+ 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  a lower
  speed.  The network  between distant  clusters might  be a  bottleneck for  the
  global performance of the application.
 
  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  a lower
  speed.  The network  between distant  clusters might  be a  bottleneck for  the
  global performance of the application.
 

-\subsection{Comparing GMRES and Multisplitting algorithms in
-synchronous mode}
+\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}

 
 

-In the scope of this paper, our first objective is to demonstrate the
-Algo-2 (Multisplitting method) shows a better performance in grid
-architecture compared with Algo-1 (Classical GMRES) both running in
-\textit{synchronous mode}. Better algorithm performance
-should means a less number of iterations output and a less execution time
-before reaching the convergence. For a systematic study, the experiments
-should figure out that, for various grid parameters values, the
-simulator will confirm the targeted outcomes, particularly for poor and
-slow networks, focusing on the impact on the communication performance
-on the chosen class of algorithm.
+In the scope  of this paper, our  first objective is to analyze  when the Krylov
+Multisplitting  method   has  better  performances  than   the  classical  GMRES
+method. With an  iterative method, better performances mean 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.

 
 The following paragraphs present the test conditions, the output results
 and our comments.\\
 
 
 
 The following paragraphs present the test conditions, the output results
 and our comments.\\
 
 

-\textit{3.a Executing the algorithms on various computational grid
+\subsubsection{Execution of the the algorithms on various computational grid

 architecture and scaling up the input matrix size}
 architecture and scaling up the input matrix size}

-\\
-
+\ \\

 % environment
 % environment

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

 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline


@@ -467,47 +490,57 @@ architecture and scaling up the input matrix size}
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
  \end{tabular}
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
  - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline
  \end{tabular}

-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}

 
 

-\end{footnotesize}

 
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 
 
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 

-In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration 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.
+In this  section, we analyze the  performences of algorithms running  on various
+grid configuration  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+show for all grid configuration 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{wrapfigure}{l}{100mm}

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

-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
-%\label{overflow}}
+  \begin{center}
+    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+  \end{center}
+  \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+  \label{fig:01}

 \end{figure}
 \end{figure}

-%\end{wrapfigure}

 
 

-The execution time difference between the two algorithms is important when
-comparing between different grid architectures, even with the same number of
-processors (like 2x16 and 4x8 = 32 processors for example). The
-experiment concludes the low sensitivity of the 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\%) less when running from 2x16=32 to 8x8=64 processors.

 
 

-\textit{\\3.b Running on two different speed cluster inter-networks\\}
+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\%) less when running from 2x16=32 to 8x8=64 processors.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Running on two different speed cluster inter-networks}
+\ \\
+
+\begin{figure} [ht!]
+\begin{center}

 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\

- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline 

  \end{tabular}
  \end{tabular}

-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
+\caption{Clusters x Nodes - Networks N1 x N2}
+\end{center}
+\end{figure}

 
 
 
 
 
 


@@ -516,31 +549,30 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Cluster x Nodes N1 x N2}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Cluster x Nodes N1 x N2}

-%\label{overflow}}
+\label{fig:02}

 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and also on a less performant network (N2).
-Figure 4 shows that end users will gain to reduce the execution time
-for both algorithms in using a grid architecture like 4x16 or 8x8: the
-performance was increased in a factor of 2. The results depict also that
-when the network speed drops down (12.5\%), the difference between the execution
-times can reach more than 25\%.
+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).
+Figure~\ref{fig:02} shows that end users will  gain to reduce the execution time
+for  both  algorithms  in using  a  grid  architecture  like  4x16 or  8x8:  the
+performance was increased  in a factor of  2. The results depict  also that when
+the  network speed  drops down  (12.5\%), the  difference between  the execution
+times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?}

 
 

-\textit{\\3.c Network latency impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline

- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline

  \end{tabular}
  \end{tabular}

-Table 3 : Network latency impact \\
-
-\end{footnotesize}
+\caption{Network latency impact}
+\end{figure}

 
 
 
 
 
 


@@ -548,123 +580,124 @@ Table 3 : Network latency impact \\
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impact on execution time}
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impact on execution time}

-%\label{overflow}}
+\label{fig:03}

 \end{figure}
 
 
 \end{figure}
 
 

-According the results in figure 5, degradation of the network
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time
-increase more than 75\% (resp. 82\%) of the execution for the classical
-GMRES (resp. multisplitting) algorithm. In addition, it appears that the
-multisplitting method tolerates more the network latency variation with
-a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-5
-}$), the execution time for GMRES is almost the double of the time for
-the multisplitting, even though, the performance was on the same order
-of magnitude with a latency of 8.10$^{-6}$.
-
-\textit{\\3.d Network bandwidth impacts on performance\\}
+According  the results  in  Figure~\ref{fig:03}, a  degradation  of the  network
+latency from 8.10$^{-6}$  to 6.10$^{-5}$ implies an absolute  time increase 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.   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$^{-6}$.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
  \end{tabular}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
  \end{tabular}

-Table 4 : Network bandwidth impact \\
-
-\end{footnotesize}
+\caption{Network bandwidth impact}
+\end{figure}

 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impact on execution time}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impact on execution time}

-%\label{overflow}
+\label{fig:04}

 \end{figure}
 
 
 
 \end{figure}
 
 
 

-The results of increasing the network bandwidth show the improvement
-of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.
-
-\textit{\\3.e Input matrix size impacts on performance\\}
+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 classical GMRES.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 4x8\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 4x8\\ %\hline

- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ 
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline

  \end{tabular}
  \end{tabular}

-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Input matrix size impact}
+\end{figure}

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

-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impact on execution time}
+\label{fig:05}

 \end{figure}
 
 \end{figure}
 

-In this experimentation, 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 the figure 7,
-the execution time for the two algorithms convergence increases with the
-input matrix size. But the interesting results here direct on (i) the
-drastic increase (300 times) of the number of iterations needed before
-the convergence for the classical GMRES algorithm when the matrix size
-go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost
-the double from N$_{x}$=140 compared with the convergence time of the
-multisplitting method. 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. Note that the
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{\\3.f CPU Power impact on performance\\}
+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 (300 times) \RC{Je ne vois pas cela sur la figure}
+of the  number of  iterations needed  to reach the  convergence for  the classical
+GMRES algorithm when  the matrix size go beyond N$_{x}$=150;
+\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.

 
 

-% environment
-\begin{footnotesize}
+\subsubsection{CPU Power impact on performance}
+
+\begin{figure} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}

-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
-
+\caption{CPU Power impact}
+\end{figure}

 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}

-%\label{overflow}}
+\label{fig:06}

 \end{figure}
 
 \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 the figure 6
-confirm the performance gain, around 95\% for both of the two methods,
-after adding more powerful CPU. 
-
-\subsection{Comparing GMRES in native synchronous mode and
-Multisplitting algorithms in asynchronous mode}
-
-The previous paragraphs put in evidence the interests to simulate the
-behavior of the application before any deployment in a real environment.
-We have focused the study on analyzing the performance in varying the
-key factors impacting the results. The study compares
-the performance of the two proposed algorithms both in \textit{synchronous mode
-}. In this section, following the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textit{
-asynchronous mode} compared with the classical GMRES staying in
-\textit{synchronous mode}.
+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.
+
+\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+
+The previous paragraphs  put in evidence the interests to  simulate the behavior
+of the application before any deployment in a real environment.  We have focused
+the study on analyzing the performance  in varying the key factors impacting the
+results. The study compares the performance  of the two proposed algorithms both
+in  \textit{synchronous mode  }. In  this section,  following the  same previous
+methodology, the  goal is  to demonstrate the  efficiency of  the multisplitting
+method in \textit{ asynchronous mode}  compared with the classical GMRES staying
+in \textit{synchronous mode}.

 
 Note that the interest of using the asynchronous mode for data exchange
 is mainly, in opposite of the synchronous mode, the non-wait aspects of
 
 Note that the interest of using the asynchronous mode for data exchange
 is mainly, in opposite of the synchronous mode, the non-wait aspects of