Typos.

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 42f4b5d37762f3602b5c5ba46962b6e31f7ca74a..0b7dc1da38fe22e09ff1b8228150999a128d563f 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -21,7 +21,6 @@
 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}
 \usepackage{algpseudocode}
 %\usepackage{amsthm}
 \usepackage{graphicx}

-\usepackage[american]{babel}

 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}


@@ -45,6 +44,8 @@
   \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]}


@@ -73,23 +74,26 @@
 analysis of simulated grid-enabled numerical iterative algorithms}
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 analysis of simulated grid-enabled numerical iterative algorithms}
 %\itshape{\journalnamelc}\footnotemark[2]}
 

-\author{    Charles Emile Ramamonjisoa and
-    David Laiymani and
-    Arnaud Giersch and
-    Lilia Ziane Khodja and
-    Raphaël Couturier
+\author{Charles Emile Ramamonjisoa\affil{1},
+    David Laiymani\affil{1},
+    Arnaud Giersch\affil{1},
+    Lilia Ziane Khodja\affil{2} and
+    Raphaël Couturier\affil{1}

 }
 
 \address{
 }
 
 \address{

-       \centering
-    Femto-ST Institute - DISC Department\\
-    Université de Franche-Comté\\
-    Belfort\\
-    Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+  \affilnum{1}%
+  Femto-ST Institute, DISC Department,
+  University of Franche-Comté,
+  Belfort, France.
+  Email:~\email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}\break
+  \affilnum{2}
+  Department of Aerospace \& Mechanical Engineering,
+  Non Linear Computational Mechanics,
+  University of Liege, Liege, Belgium.
+  Email:~\email{l.zianekhodja@ulg.ac.be}

 }
 
 }
 

-%% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. Email: l.zianekhodja@ulg.ac.be
-

 \begin{abstract}   The behavior of multi-core applications is always a challenge
 to predict, especially with a new architecture for which no experiment has been
 performed. With some applications, it is difficult, if not impossible, to build
 \begin{abstract}   The behavior of multi-core applications is always a challenge
 to predict, especially with a new architecture for which no experiment has been
 performed. With some applications, it is difficult, if not impossible, to build


@@ -132,7 +136,7 @@ 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\dots{} 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
 
 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


@@ -172,10 +176,22 @@ applications (i.e. large linear system solvers) can help developers to better
 tune their application for a given multi-core architecture. To show the validity
 of this approach we first compare the simulated execution of the multisplitting
 algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
 tune their application for a given multi-core architecture. To show the validity
 of this approach we first compare the simulated execution of the multisplitting
 algorithm  with  the  GMRES   (Generalized   Minimal  Residual)

-solver~\cite{saad86} in synchronous mode. The obtained results on different
+solver~\cite{saad86} in synchronous mode. 
+
+\LZK{Pas trop convainquant comme argument pour valider l'approche de simulation. \\On peut dire par exemple: on a pu simuler différents algos itératifs à large échelle (le plus connu GMRES et deux variantes de multisplitting) et la simulation nous a permis (sans avoir le vrai matériel) de déterminer quelle serait la meilleure solution pour une telle configuration de l'archi ou vice versa.\\A revoir...}
+
+The obtained results on different

 simulated multi-core architectures confirm the real results previously obtained
 simulated multi-core architectures confirm the real results previously obtained

-on non simulated architectures.  We also confirm  the efficiency  of the
-asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. In
+on non simulated architectures.  
+
+\LZK{Il n y a pas dans la partie expé cette comparaison et confirmation des résultats entre la simulation et l'exécution réelle des algos sur les vrais clusters.\\ Sinon on pourrait ajouter dans la partie expé une référence vers le journal supercomput de krylov multi pour confirmer que cette méthode est meilleure que GMRES sur les clusters large échelle.}
+
+We also confirm  the efficiency  of the
+asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. 
+
+\LZK{P.S.: Pour tout le papier, le principal objectif n'est pas de faire des comparaisons entre des méthodes itératives!!\\Sinon, les deux algorithmes Krylov multisplitting synchrone et multisplitting asynchrone sont plus efficaces que GMRES sur des clusters à large échelle.\\Et préciser, si c'est vraiment le cas, que le multisplitting asynchrone est plus efficace et adapté aux clusters distants par rapport aux deux autres algos (je n'ai pas encore lu la partie expé)}
+
+In

 this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
 this way and with a simple computing architecture (a laptop) SimGrid allows us
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no


@@ -189,8 +205,10 @@ Section~\ref{sec:04} details the different solvers that we use.  Finally our
 experimental results are presented in section~\ref{sec:expe} followed by some
 concluding remarks and perspectives.
 
 experimental results are presented in section~\ref{sec:expe} followed by some
 concluding remarks and perspectives.
 

+\LZK{Proposition d'un titre pour le papier: Grid-enabled simulation of large-scale linear iterative solvers.}
+

 
 

-\section{The asynchronous iteration model}
+\section{The asynchronous iteration model and the motivations of our work}

 \label{sec:asynchro}
 
 Asynchronous iterative methods have been  studied for many years theoritecally and
 \label{sec:asynchro}
 
 Asynchronous iterative methods have been  studied for many years theoritecally and


@@ -214,8 +232,24 @@ point. In the  asynchronous model, the convergence detection is  more tricky as
 it   must  not   synchronize  all   the  processors.   Interested  readers   can
 consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 
 it   must  not   synchronize  all   the  processors.   Interested  readers   can
 consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
 

+The number of iterations required to reach the convergence is generally greater
+for the asynchronous scheme (this number depends depends on  the delay of the
+messages). Note that, it is not the case in the synchronous mode where the
+number of iterations is the same than in the sequential mode. In this way, the
+set of the parameters  of the  platform (number  of nodes,  power of nodes,
+inter and  intra clusters  bandwidth  and  latency, \ldots) and  of  the
+application can drastically change the number of iterations required to get the
+convergence. It follows that asynchronous iterative algorithms are difficult to
+optimize since the financial and deployment costs on large scale multi-core
+architecture are often very important. So, prior to delpoyment and tests it
+seems very promising to be able to simulate the behavior of asynchronous
+iterative algorithms. The problematic is then to show that the results produce
+by simulation are in accordance with reality i.e. of the same order of
+magnitude. To our knowledge, there is no study on this problematic.
+

 \section{SimGrid}
 \section{SimGrid}

- \label{sec:simgrid}
+\label{sec:simgrid}
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile} is a discrete event simulation framework to study the behavior of large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds and High Performance Computation systems. It is widely used to simulate and evaluate heuristics, prototype applications or even assess legacy MPI applications. It is still actively developed by the scientific community and distributed as an open source software.

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


@@ -344,18 +378,16 @@ 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}
 
 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.
 \end{itemize}
 
 It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.


@@ -432,13 +464,13 @@ 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}


@@ -464,8 +496,8 @@ 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 a synchronous  iterative method, better performances mean a
+Multisplitting  method   has  better  performance  than   the  classical  GMRES
+method. With a synchronous  iterative method, better performance means a

 smaller number of iterations and execution time before reaching the convergence.
 For a systematic study,  the experiments  should figure  out  that, for  various
 grid  parameters values, the simulator will confirm  the targeted outcomes,
 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,


@@ -481,26 +513,25 @@ 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{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}}
+\label{tab:01}

 \end{center}
 \end{center}

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

 
 
 
 
 
 

-%\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
+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
 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


@@ -514,7 +545,7 @@ 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\RC{idem}}

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


@@ -524,56 +555,57 @@ 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. $48\%$) when running from 2x16=32 to 8x8=64 processors.
+$40\%$ 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?}

 
 

-\subsubsection{Running on two different inter-clusters network speed}
-\ \\
+\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$^{-5}$ \\
  Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
  \end{tabular}
  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
  \end{tabular}

-\caption{Clusters x Nodes - Networks N1 x N2}
+\caption{Test conditions: grid 2x16 and 4x8 with  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). \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\%.
+%\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 with 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  by 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?}
-\DL{pas clair}

 
 \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 impacts}
-\end{figure}
+\caption{Test conditions: network latency impacts}
+\label{tab:03}
+\end{table}

 
 
 
 
 
 


@@ -585,28 +617,30 @@ times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi
 \end{figure}
 
 
 \end{figure}
 
 

-According to the results  of  Figure~\ref{fig:03}, a  degradation  of the  network
-latency from $8.10^{-6}$  to $6.10^{-5}$ implies an absolute  time increase of more
-than $75\%$  (resp. $82\%$) of the  execution for the classical  GMRES (resp. Krylov
-multisplitting)   algorithm.   In   addition,   it  appears   that  the   Krylov
-multisplitting method tolerates  more the network latency variation  with a less
-rate  increase  of  the  execution   time.   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 $8.10^{-6}$.
+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}
 \ \\
 
 \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 impacts}
-\end{figure}
+\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!]


@@ -620,39 +654,40 @@ 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
 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
+ Grid Architecture & 4x8\\ %\hline

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

-\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)  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}
 
   compared with the Krylov multisplitting method.
 \end{enumerate}
 


@@ -661,79 +696,87 @@ 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
 \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
-the current computation after a communication operation like sending
-some data between nodes. Each processor can continue their local
-calculation without waiting for the end of the communication. Thus, the
-asynchronous may theoretically reduce the overall execution time and can
-improve the algorithm performance.
-
-As stated supra, Simgrid simulator tool has been used to prove 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 : \\
+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} compared with the
+classical GMRES in \textit{synchronous mode}.

 
 

-% environment
-\begin{footnotesize}
+The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
+synchronization. With  geographically distant  clusters, this may  be essential.
+In  this case,  each  processor can  compute its  iteration  freely without  any
+synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
+theoretically reduce  the overall execution  time and can improve  the algorithm
+performance.
+
+\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
+In this section, Simgrid simulator tool has been successfully used to show
+the efficiency of  the multisplitting in asynchronous mode and  to find the best
+combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to
+get    the   highest    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
+exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+
+
+The test conditions are summarized in the table~\ref{tab:07}: \\
+
+\begin{table} [ht!]
+\centering

 \begin{tabular}{r c }
  \hline
 \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{footnotesize}
+\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 varying the
-CPU power and the network parameters (bandwidth and latency) in the
-simulator tool with different problem size. The relative gains greater
-than 1 between the two algorithms have been captured after each step of
-the test. Table 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
+Again,  comprehensive and  extensive tests  have been  conducted with  different
+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
+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}}


@@ -744,14 +787,12 @@ the classical GMRES execution and convergence time. The experimentation has demo
     \end{tabular}}
 
 
     \end{tabular}}
 
 

-\begin{table}[!t]
-  \centering
+\begin{figure}[!t]
+\centering
+%\begin{table}

 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 %  \label{"Table 7"}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 %  \label{"Table 7"}

-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
-  \begin{mytable}{11}
+ \begin{mytable}{11}

     \hline
     bandwidth (Mbit/s)
     & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\
     \hline
     bandwidth (Mbit/s)
     & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\


@@ -772,17 +813,20 @@ the classical GMRES \\
     & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
     \hline
   \end{mytable}
     & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
     \hline
   \end{mytable}

-\end{table}
+%\end{table}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{fig:07}
+\end{figure}
+

 
 \section{Conclusion}
 CONCLUSION
 
 
 
 \section{Conclusion}
 CONCLUSION
 
 

-\section*{Acknowledgment}
-
+%\section*{Acknowledgment}
+\ack

 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 
 This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 

-

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