RCE: Modification des labels des tableaux et des graphiques

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 82777e31dc256726f9b307977e12fda64c3610c0..34f7ec75f8636c2ae3278b1e157868fb86a8ffaa 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -45,6 +45,8 @@
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RCE}[2][inline]{%
   \todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RCE}[2][inline]{%
   \todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}

+\newcommand{\DL}[2][inline]{%
+    \todo[color=pink!10,#1]{\sffamily\textbf{DL:} #2}\xspace}

 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}
 
 \algnewcommand\algorithmicinput{\textbf{Input:}}
 \algnewcommand\Input{\item[\algorithmicinput]}


@@ -96,20 +98,21 @@ performed. With some applications, it is difficult, if not impossible, to build
 accurate performance models. That is why another solution is to use a simulation
 tool which allows us to change many parameters of the architecture (network
 bandwidth, latency, number of processors) and to simulate the execution of such
 accurate performance models. That is why another solution is to use a simulation
 tool which allows us to change many parameters of the architecture (network
 bandwidth, latency, number of processors) and to simulate the execution of such

-applications. We have decided to use SimGrid as it enables to benchmark MPI
-applications.
+applications. The main contribution of this paper is to show that the use of a
+simulation tool (here we have decided to use the SimGrid toolkit) can really
+help developpers to better tune their applications for a given multi-core
+architecture.

 
 

-In this paper, we focus our attention on two parallel iterative algorithms based
+In particular we focus our attention on two parallel iterative algorithms based

 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
 on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.

-These algorithms  are used to  solve libear  systems. Two different  variants of
+These algorithms  are used to  solve linear  systems. Two different  variants of

 the Multisplitting are studied: one  using synchronoous  iterations and  another
 the Multisplitting are studied: one  using synchronoous  iterations and  another

-one  with asynchronous iterations. For each algorithm we have  tested different
-parameters to see their influence.  We strongly  recommend people  interested
-by investing  into a  new expensive  hardware  architecture  to   benchmark
-their  applications  using  a simulation tool before.
-
-
-
+one  with asynchronous iterations. For each algorithm we have simulated
+different architecture parameters to evaluate their influence on the overall
+execution time.  The obtain simulated results confirm the real results
+previously obtained on different real multi-core architectures and also confirm
+the efficiency of the asynchronous multisplitting algorithm compared to the
+synchronous GMRES method.

 
 \end{abstract}
 
 
 \end{abstract}
 


@@ -238,7 +241,7 @@ 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 ({\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{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]


@@ -343,19 +346,19 @@ nodes/processors for each cluster).
 In addition, the following arguments are given to the programs at runtime:
 
 \begin{itemize}
 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;
-       \item maximum number of the GMRES restarts in the Arnorldi process;
-       \item maximum number of iterations and 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 is fixed to $6.0$ 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.
+       \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 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 maximum number of iterations and precision for the classical GMRES method,
+        \item maximum number of restarts for the Arnorldi process in GMRES method,
+       \item execution mode: synchronous or asynchronous.

 \end{itemize}
 \end{itemize}

+\LZK{CE pourrais tu vérifier et confirmer les valeurs des éléments diag et off-diag de la matrice?}
+\RCE{oui, les valeurs de diag et off-diag donnees sont ok}

 
 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.
 


@@ -431,13 +434,13 @@ input data.  \\
 a grid environment}
 
 When running a distributed application in a computational grid, many factors may
 a grid environment}
 
 When running a distributed application in a computational grid, many factors may

-have a strong impact on the performances.  First of all, the architecture of the
+have a strong impact on the performance.  First of all, the architecture of the

 grid itself can obviously influence the  performance results of the program. The
 performance gain  might be important  theoretically when the number  of clusters
 and/or  the  number  of  nodes (processors/cores)  in  each  individual  cluster
 increase.
 
 grid itself can obviously influence the  performance results of the program. The
 performance gain  might be important  theoretically when the number  of clusters
 and/or  the  number  of  nodes (processors/cores)  in  each  individual  cluster
 increase.
 

-Another important factor  impacting the overall performances  of the application
+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}


@@ -463,35 +466,36 @@ and  between distant  clusters.  This parameter is application dependent.
 \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
 \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  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.
+Multisplitting  method   has  better  performance  than   the  classical  GMRES
+method. With a synchronous  iterative method, better performance 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.\\
 
 

-\subsubsection{Execution of the the algorithms on various computational grid
-architecture and scaling up the input matrix size}
+\subsubsection{Execution of the algorithms on various computational grid
+architectures and scaling up the input matrix size}

 \ \\
 % environment
 
 \ \\
 % environment
 

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

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

- Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
+ Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline

  Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  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}
  Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  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}

-\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)}}
+\caption{Test conditions: Various grid configurations with the input matix size 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)}}
+\label{tab:01}

 \end{center}
 \end{center}

-\end{figure}
+\end{table}

 
 
 
 
 
 


@@ -499,10 +503,10 @@ architecture and scaling up the input matrix size}
 %\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 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
+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.
 
 \RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
 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...}


@@ -513,7 +517,7 @@ multisplitting method.
   \begin{center}
     \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
   \end{center}
   \begin{center}
     \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
   \end{center}

-  \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+  \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170}

   \label{fig:01}
 \end{figure}
 
   \label{fig:01}
 \end{figure}
 


@@ -523,136 +527,139 @@ 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
 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.
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.

 
 

-\subsubsection{Running on two different speed cluster inter-networks}
-\ \\
+\subsubsection{Running on two different inter-clusters network speeds \\} 

 
 

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

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

- Grid & 2x16, 4x8\\ %\hline
+ Grid Architecture & 2x16, 4x8\\ %\hline

  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  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}

-\caption{Clusters x Nodes - Networks N1 x N2}
+\caption{Test conditions: Grid 2x16 and 4x8 - Networks N1 vs N2}
+\label{tab:02}

 \end{center}
 \end{center}

-\end{figure}
+\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).
+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  by a factor of  $2$. 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\%. 
+%\RC{c'est pas clair : la différence entre quoi et quoi?}
+%\DL{pas clair}
+%\RCE{Modifie}

 
 
 %\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 
 
 %\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Cluster x Nodes N1 x N2}
+\caption{Grid 2x16 and 4x8 - Networks N1 vs N2}

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

-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?}

 
 \subsubsection{Network latency impacts on performance}
 \ \\
 
 \subsubsection{Network latency impacts on performance}
 \ \\

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

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

- Grid & 2x16\\ %\hline
+ Grid Architecture & 2x16\\ %\hline

  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
  \end{tabular}
  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
  \end{tabular}

-\caption{Network latency impact}
-\end{figure}
+\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}
 
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}

-\caption{Network latency impact on execution time}
+\caption{Network latency impacts on execution time}

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

-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
+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.   Consequently,  in  the  worst  case
 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
+($lat=6.10^{-5 }$), the  execution time for GMRES is almost  the double than the

 time of the Krylov multisplitting, even  though, the performance was on the same
 time of the Krylov multisplitting, even  though, the performance was on the same

-order of magnitude with a latency of 8.10$^{-6}$.
+order of magnitude with a latency of $8.10^{-6}$.

 
 \subsubsection{Network bandwidth impacts on performance}
 \ \\
 
 \subsubsection{Network bandwidth impacts on performance}
 \ \\

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

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

- Grid & 2x16\\ %\hline
+ Grid Architecture & 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}
  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}

-\caption{Network bandwidth impact}
-\end{figure}
+\caption{Test conditions: Network bandwidth impacts}
+\label{tab:04}
+\end{table}

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

-\caption{Network bandwith impact on execution time}
+\caption{Network bandwith impacts on execution time}

 \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
 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.
+of $40\%$ which is only around $24\%$ for the classical GMRES.

 
 \subsubsection{Input matrix size impacts on performance}
 \ \\
 
 \subsubsection{Input matrix size impacts on performance}
 \ \\

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

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

- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ 
+ Grid Architecture & 4x8\\ %\hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\

  Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
  Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
  \end{tabular}

-\caption{Input matrix size impact}
-\end{figure}
+\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}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}

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

 \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
+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}
 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
+  \item the drastic increase ($10$ times) \RC{Je ne vois pas cela sur la figure}
+\RCE{Corrige} 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}
 
   compared with the Krylov multisplitting method.
 \end{enumerate}
 


@@ -661,38 +668,44 @@ 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.
 
 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 impact on performance}
+\subsubsection{CPU Power impacts on performance}

 
 

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

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

- Grid & 2x16\\ %\hline
+ Grid architecture & 2x16\\ %\hline

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

-\caption{CPU Power impact}
-\end{figure}
+\caption{Test conditions: CPU Power impacts}
+\label{tab:06}
+\end{table}

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

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

 \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
 \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
+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.
 
 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}
+
+

 \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.  In  this
 section, following  the same previous  methodology, our  goal is to  compare the
 \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.  In  this
 section, following  the same previous  methodology, our  goal is to  compare the

-efficiency of the multisplitting method  in \textit{ asynchronous mode} with the
+efficiency of the multisplitting method  in \textit{ asynchronous mode} compared with the

 classical GMRES in \textit{synchronous mode}.
 
 The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
 classical GMRES in \textit{synchronous mode}.
 
 The  interest of  using  an asynchronous  algorithm  is that  there  is no  more


@@ -703,37 +716,39 @@ 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}
 performance.
 
 \RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}

-As stated before, the Simgrid simulator tool has been successfully used to show
+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.
 
 
 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.
 
 

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

 
 

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

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

- Grid & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2x50 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$^{-2}$\\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
  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$^{-2}$\\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}

-\end{figure}
+\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
+\label{tab:07}
+\end{table}

 
 Again,  comprehensive and  extensive tests  have been  conducted with  different
 
 Again,  comprehensive and  extensive tests  have been  conducted with  different

-parametes as  the CPU power, the  network parameters (bandwidth and  latency) in
-the simulator tool  and with different problem size. The  relative gains greater
-than 1  between the  two algorithms have  been captured after  each step  of the
-test.   In  Figure~\ref{table:01}  are  reported the  best  grid  configurations
-allowing the  multisplitting method to  be more than  2.5 times faster  than the
+parameters as  the CPU power, the  network parameters (bandwidth and  latency)
+and with different problem size. The  relative gains greater than $1$  between the
+two algorithms have  been captured after  each step  of the test.   In
+Figure~\ref{fig:07}  are  reported the  best  grid  configurations allowing
+the  multisplitting method to  be more than  $2.5$ times faster  than the

 classical  GMRES.  These  experiments also  show the  relative tolerance  of the
 multisplitting algorithm when using a low speed network as usually observed with
 classical  GMRES.  These  experiments also  show the  relative tolerance  of the
 multisplitting algorithm when using a low speed network as usually observed with

-geographically distant clusters throuth the internet.
+geographically distant clusters through the internet.

 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
 
 % use the same column width for the following three tables
 \newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}


@@ -772,7 +787,7 @@ geographically distant clusters throuth the internet.
   \end{mytable}
 %\end{table}
  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
   \end{mytable}
 %\end{table}
  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}

- \label{table:01}
+ \label{fig:07}

 \end{figure}
 
 
 \end{figure}