DL : conclu. Manque les futurs works

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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+++ 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}


@@ -71,27 +70,30 @@
 
 
 
 
 
 

-\begin{document} \RCE{Titre a confirmer.} \title{Comparative performance
-analysis of simulated grid-enabled numerical iterative algorithms}
+\begin{document}
+\title{Grid-enabled simulation of large-scale linear iterative solvers}

 %\itshape{\journalnamelc}\footnotemark[2]}
 
 %\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


@@ -134,7 +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


@@ -163,22 +165,30 @@ application  on   a  given   multi-core  architecture.  Finding   good  resource
 allocations policies under  varying CPU power, network speeds and  loads is very
 challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
 problematic is  even more difficult  for the  asynchronous scheme where  a small
 allocations policies under  varying CPU power, network speeds and  loads is very
 challenging and  labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
 problematic is  even more difficult  for the  asynchronous scheme where  a small

-parameter variation of the execution platform can lead to very different numbers
-of iterations to reach the converge and so to very different execution times. In
-this challenging context we think that the  use of a simulation tool can greatly
-leverage the possibility of testing various platform scenarios.
-
-The main contribution of this paper is to show that the use of a simulation tool
-(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real  parallel
-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)
-solver~\cite{saad86} in synchronous mode. The obtained results on different
-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
-this way and with a simple computing architecture (a laptop) SimGrid allows us
+parameter variation of the execution platform and of the application data can
+lead to very different numbers of iterations to reach the converge and so to
+very different execution times. In this challenging context we think that the
+use of a simulation tool can greatly leverage the possibility of testing various
+platform scenarios.
+
+The  {\bf main  contribution  of  this paper}  is  to show  that  the  use of  a
+simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real
+parallel 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 Krylov
+multisplitting  algorithm   with  the   GMRES  (Generalized   Minimal  Residual)
+solver~\cite{saad86} in  synchronous mode.  The simulation  results allow  us to
+determine  which method  to choose  given a  specified multi-core  architecture.
+Moreover the  obtained results  on different simulated  multi-core architectures
+confirm the  real results  previously obtained  on non  simulated architectures.
+More precisely the simulated results are in accordance (i.e. with the same order
+of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
+the synchronous  multisplitting method  is more efficient  than GMRES  for large
+scale  clusters.   Simulated   results  also  confirm  the   efficiency  of  the
+asynchronous  multisplitting   algorithm  compared  to  the   synchronous  GMRES
+especially in case of geographically distant clusters.
+
+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
 related work on the large-scale multi-core simulation of a real synchronous and
 to run a test campaign  of  a  real parallel iterative  applications on
 different simulated multi-core architectures.  To our knowledge, there is no
 related work on the large-scale multi-core simulation of a real synchronous and


@@ -192,7 +202,7 @@ experimental results are presented in section~\ref{sec:expe} followed by some
 concluding remarks and perspectives.
 
 
 concluding remarks and perspectives.
 
 

-\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


@@ -216,10 +226,77 @@ 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.

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

+% SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
+% is a simulation framework to study the behavior of large-scale distributed
+% systems.  As its name suggests, it emanates from the grid computing community,
+% but is nowadays used to study grids, clouds, HPC or peer-to-peer systems.  The
+% early versions of SimGrid date back from 1999, but it is still actively
+% developed and distributed as an open source software.  Today, it is one of the
+% major generic tools in the field of simulation for large-scale distributed
+% systems.
+
+SimGrid provides several programming interfaces: MSG to simulate Concurrent
+Sequential Processes, SimDAG to simulate DAGs of (parallel) tasks, and SMPI to
+run real applications written in MPI~\cite{MPI}.  Apart from the native C
+interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
+languages.  SMPI is the interface that has been used for the work described in
+this paper.  The SMPI interface implements about \np[\%]{80} of the MPI 2.0
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications (cf Section IV - paragraph B).
+
+Within SimGrid, the execution of a distributed application is simulated by a
+single process.  The application code is really executed, but some operations,
+like communications, are intercepted, and their running time is computed
+according to the characteristics of the simulated execution platform.  The
+description of this target platform is given as an input for the execution, by
+means of an XML file.  It describes the properties of the platform, such as
+the computing nodes with their computing power, the interconnection links with
+their bandwidth and latency, and the routing strategy.  The scheduling of the
+simulated processes, as well as the simulated running time of the application
+are computed according to these properties.
+
+To compute the durations of the operations in the simulated world, and to take
+into account resource sharing (e.g. bandwidth sharing between competing
+communications), SimGrid uses a fluid model.  This allows users to run relatively fast
+simulations, while still keeping accurate
+results~\cite{bedaride+degomme+genaud+al.2013.toward,
+  velho+schnorr+casanova+al.2013.validity}.  Moreover, depending on the
+simulated application, SimGrid/SMPI allows to skip long lasting computations and
+to only take their duration into account.  When the real computations cannot be
+skipped, but the results are unimportant for the simulation results, it is
+also possible to share dynamically allocated data structures between
+several simulated processes, and thus to reduce the whole memory consumption.
+These two techniques can help to run simulations on a very large scale.
+
+The validity of simulations with SimGrid has been asserted by several studies.
+See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+referenced therein for the validity of the network models.  Comparisons between
+real execution of MPI applications on the one hand, and their simulation with
+SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+  clauss+stillwell+genaud+al.2011.single,
+  bedaride+degomme+genaud+al.2013.toward}.  All these works conclude that
+SimGrid is able to simulate pretty accurately the real behavior of the
+applications.

 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Two-stage multisplitting methods}
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Two-stage multisplitting methods}


@@ -346,18 +423,16 @@ 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.


@@ -434,13 +509,13 @@ 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}


@@ -466,8 +541,8 @@ 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,


@@ -483,26 +558,26 @@ 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?}
+\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
+\label{tab:01}

 \end{center}
 \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


@@ -516,7 +591,8 @@ 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}
+\AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}

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


@@ -526,23 +602,29 @@ 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\%.

 
 
 
 
 
 


@@ -550,71 +632,69 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors.
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Cluster x Nodes N1 x N2}
+\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}

 \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}

 
 
 
 \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 impacts on execution time}
+\caption{Network latency impacts on execution time
+\AG{\np{E-6}}}

 \label{fig:03}
 \end{figure}
 
 
 \label{fig:03}
 \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!]
 \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 impacts on execution time}
+\caption{Network bandwith impacts on execution time
+\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}

 \label{fig:04}
 \end{figure}
 
 \label{fig:04}
 \end{figure}
 


@@ -626,16 +706,17 @@ 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 impacts}
-\end{figure}
+\caption{Test conditions: Input matrix size impacts}
+\label{tab:05}
+\end{table}

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


@@ -651,9 +732,9 @@ In these experiments, the input matrix size  has been set from $N_{x} = N_{y}
 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 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}
 \item the  classical GMRES execution time  is almost the double  for $N_{x}=140$
   compared with the Krylov multisplitting method.
 \end{enumerate}


@@ -665,16 +746,17 @@ grid 2x16 leading to the same conclusion.
 
 \subsubsection{CPU Power impacts 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 impacts}
-\end{figure}
+\caption{Test conditions: CPU Power impacts}
+\label{tab:06}
+\end{table}

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


@@ -688,17 +770,24 @@ 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.
 from $1$ to $19$ GFlops.  The outputs  depicted in Figure~\ref{fig:06}  confirm the
 performance gain,  around $95\%$ for  both of the  two methods, after  adding more
 powerful CPU.

+\ \\
+%\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
+%obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
+%besoin de déployer sur une archi réelle}
+
+To conclude these series of experiments, with  SimGrid we have been able to make
+many simulations  with many parameters  variations. Doing all  these experiments
+with a real platform is most of  the time not possible. Moreover the behavior of
+both GMRES and  Krylov multisplitting methods is in accordance  with larger real
+executions on large scale supercomputer~\cite{couturier15}.

 
 

-\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


@@ -709,37 +798,39 @@ 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}}


@@ -777,23 +868,46 @@ geographically distant clusters throuth the internet.
     \hline
   \end{mytable}
 %\end{table}
     \hline
   \end{mytable}
 %\end{table}

- \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
- \label{table:01}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES
+\AG{C'est un tableau, pas une figure}}
+ \label{fig:07}

 \end{figure}
 
 
 \section{Conclusion}
 \end{figure}
 
 
 \section{Conclusion}

-CONCLUSION

 
 

+In this paper we have presented the simulation of the execution of three
+different parallel solvers on some multi-core architectures. We have show that
+the SimGrid toolkit is an interesting simulation tool that has allowed us to
+determine  which method  to choose  given a  specified multi-core  architecture.
+Moreover the simulated results are in accordance (i.e. with the same order of
+magnitude)  with the works  presented in~\cite{couturier15}. Simulated   results
+also  confirm  the   efficiency  of  the asynchronous  multisplitting
+algorithm  compared  to  the   synchronous  GMRES especially in case of
+geographically distant clusters.

 
 

-\section*{Acknowledgment}
+These results are important since it is very  time consuming to find optimal
+configuration  and deployment requirements for a given application  on   a given
+multi-core  architecture. Finding   good  resource allocations policies under
+varying CPU power, network speeds and  loads is very challenging and  labor
+intensive. This problematic is  even more difficult  for the  asynchronous
+scheme where  a small parameter variation of the execution platform and of the
+application data can lead to very different numbers of iterations to reach the
+converge and so to very different execution times.
+
+
+Our future works...

 
 

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

 
 
 
 

+%\section*{Acknowledgment}
+\ack
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
+

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+

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