Corrections coquilles sec 02

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index a21da9ab5eb292cc582c0fb627abecb36514724e..7fd9704c72ff315cc3e13f9ff24459a5e1905d38 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -70,8 +70,8 @@
 
 
 
 
 
 

-\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]}
 
 \author{Charles Emile Ramamonjisoa\affil{1},
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{Charles Emile Ramamonjisoa\affil{1},


@@ -94,27 +94,26 @@ analysis of simulated grid-enabled numerical iterative algorithms}
   Email:~\email{l.zianekhodja@ulg.ac.be}
 }
 
   Email:~\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
-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. 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 particular we focus our attention on two parallel iterative algorithms based
-on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
-These algorithms  are used to  solve linear  systems. Two different  variants of
-the Multisplitting are studied: one  using synchronoous  iterations and  another
-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.
+\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
+%% 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. 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 developers to better tune their applications for a given multi-core
+%% architecture.
+
+%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations.  
+%% For each algorithm we have simulated
+%% different architecture parameters to evaluate their influence on the overall
+%% execution time. 
+%% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
+
+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 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. 
+
+In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.

 
 \end{abstract}
 
 
 \end{abstract}
 


@@ -150,10 +149,10 @@ task cannot begin a new iteration while it has not received data dependencies
 from its neighbors. We say that the iteration computation follows a
 \textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
 iteration without having to wait for the data dependencies coming from its
 from its neighbors. We say that the iteration computation follows a
 \textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
 iteration without having to wait for the data dependencies coming from its

-neighbors. Both communication and computations are \textit{asynchronous}
+neighbors. Both communications and computations are \textit{asynchronous}

 inducing that there is no more idle time, due to synchronizations, between two
 iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
 inducing that there is no more idle time, due to synchronizations, between two
 iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks

-that we detail in section~\ref{sec:asynchro} but even if the number of
+that we detail in Section~\ref{sec:asynchro} but even if the number of

 iterations required to converge is generally  greater  than for the synchronous
 case, it appears that the asynchronous  iterative scheme  can significantly
 reduce  overall execution times by  suppressing idle  times due to
 iterations required to converge is generally  greater  than for the synchronous
 case, it appears that the asynchronous  iterative scheme  can significantly
 reduce  overall execution times by  suppressing idle  times due to


@@ -166,39 +165,29 @@ 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 and of the application data can
 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 and of the application data can

-lead to very different numbers of iterations to reach the converge and so to
+lead to very different numbers of iterations to reach the convergence 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.
 
 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 simulation results allow us to
-determine which method to choose given a specified multi-core architecture.
-
-\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...}
-\DL{OK : ajout d'une phrase précisant tout cela}
-
-Moreover the obtained results on different simulated multi-core architectures
-confirm the real results previously obtained on non simulated architectures.
+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  applications 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  for a given 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
 More precisely the simulated results are in accordance (i.e. with the same order

-of magnitude) with the works presented in [], which show that the multisplitting
-method is more efficient than GMRES for large scale clusters.
+of magnitude)  with the works  presented in~\cite{couturier15}, which  show that
+the synchronous  Krylov 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.

 
 

-\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.}
-\DL{OK ajout d'une phrase. Par contre je n'ai pas la ref. Merci de la mettre}
-
-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
+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


@@ -206,22 +195,20 @@ asynchronous iterative application.
 
 This paper is organized as follows. Section~\ref{sec:asynchro} presents the
 iteration model we use and more particularly the asynchronous scheme.  In
 
 This paper is organized as follows. Section~\ref{sec:asynchro} presents the
 iteration model we use and more particularly the asynchronous scheme.  In

-section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
+Section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.

 Section~\ref{sec:04} details the different solvers that we use.  Finally our
 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
+experimental results are presented in Section~\ref{sec:expe} followed by some

 concluding remarks and perspectives.
 
 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 and the motivations of our work}
 \label{sec:asynchro}
 
 
 \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
+Asynchronous iterative methods have been  studied for many years theoretically and

 practically. Many methods have been considered and convergence results have been
 proved. These  methods can  be used  to solve, in  parallel, fixed  point problems
 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,
 practically. Many methods have been considered and convergence results have been
 proved. These  methods can  be used  to solve, in  parallel, fixed  point problems
 (i.e. problems  for which  the solution is  $x^\star =f(x^\star)$.  In practice,

-asynchronous iterations  methods can be used  to solve, for example,  linear and
+asynchronous iteration  methods can be used  to solve, for example,  linear and

 non-linear systems of equations or optimization problems, interested readers are
 invited to read~\cite{BT89,bahi07}.
 
 non-linear systems of equations or optimization problems, interested readers are
 invited to read~\cite{BT89,bahi07}.
 


@@ -231,7 +218,7 @@ algorithm that supports both the synchronous or the asynchronous iteration model
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In
 the synchronous  mode, iterations are  synchronized whereas in  the asynchronous
 requires very few modifications  to be able to be executed  in both variants. In
 practice, only  the communications and  convergence detection are  different. In
 the synchronous  mode, iterations are  synchronized whereas in  the asynchronous

-one, they are not.  It should be noticed that non blocking communications can be
+one, they are not.  It should be noticed that non-blocking communications can be

 used in both  modes. Concerning the convergence  detection, synchronous variants
 can use  a global convergence procedure  which acts as a  global synchronization
 point. In the  asynchronous model, the convergence detection is  more tricky as
 used in both  modes. Concerning the convergence  detection, synchronous variants
 can use  a global convergence procedure  which acts as a  global synchronization
 point. In the  asynchronous model, the convergence detection is  more tricky as


@@ -239,17 +226,17 @@ 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
 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
+for the asynchronous scheme (this number 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,
 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
+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
 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
+architectures are often very important. So, prior to deployment and tests it

 seems very promising to be able to simulate the behavior of asynchronous
 seems very promising to be able to simulate the behavior of asynchronous

-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced

 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.
 
 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.
 


@@ -576,8 +563,8 @@ architectures and scaling up the input matrix size}
  \hline
  Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
  Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  \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
+ Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
+ - &  N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$  =170 $\times$ 170 $\times$ 170    \\ \hline

  \end{tabular}
 \caption{Test conditions: various grid configurations with the 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.}}
  \end{tabular}
 \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.}}


@@ -603,7 +590,7 @@ 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{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170\RC{idem}
+  \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}
 \AG{Utiliser le point comme séparateur décimal et non la virgule.  Idem dans les autres figures.}}
   \label{fig:01}
 \end{figure}


@@ -625,7 +612,7 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC
  Grid Architecture & 2x16, 4x8\\ %\hline
  Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  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
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline

  \end{tabular}
 \caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
 \label{tab:02}
  \end{tabular}
 \caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
 \label{tab:02}


@@ -637,9 +624,7 @@ speed inter-cluster  network (N1) and  also on  a less performant  network (N2).
 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\%.
 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{wrapfigure}{l}{100mm}


@@ -661,7 +646,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
  \hline
  Grid Architecture & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  \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
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline

  \end{tabular}
 \caption{Test conditions: network latency impacts}
 \label{tab:03}
  \end{tabular}
 \caption{Test conditions: network latency impacts}
 \label{tab:03}


@@ -697,7 +682,7 @@ magnitude with a latency of $8.10^{-6}$.
  \hline
  Grid Architecture & 2x16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  \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 \\
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\

  \end{tabular}
 \caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
 \label{tab:04}
  \end{tabular}
 \caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
 \label{tab:04}


@@ -726,7 +711,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
  \hline
  Grid Architecture & 4x8\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
  \hline
  Grid Architecture & 4x8\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\

- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
+ Input matrix size & $N_{x}$ = From 40 to 200\\ \hline

  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}
  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}


@@ -766,7 +751,7 @@ grid 2x16 leading to the same conclusion.
  \hline
  Grid architecture & 2x16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  \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
+ Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline

  \end{tabular}
 \caption{Test conditions: CPU Power impacts}
 \label{tab:06}
  \end{tabular}
 \caption{Test conditions: CPU Power impacts}
 \label{tab:06}


@@ -784,10 +769,16 @@ 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}

 
 

-\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}.

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


@@ -823,7 +814,7 @@ The test conditions are summarized in the table~\ref{tab:07}: \\
  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\

- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
+ Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline

  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
 \caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
  Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
 \caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}


@@ -834,7 +825,7 @@ 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
 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
+Table~\ref{tab:08}  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
 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


@@ -849,7 +840,7 @@ geographically distant clusters through the internet.
     \end{tabular}}
 
 
     \end{tabular}}
 
 

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

 \centering
 %\begin{table}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
 \centering
 %\begin{table}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}


@@ -876,14 +867,39 @@ geographically distant clusters through the internet.
     \hline
   \end{mytable}
 %\end{table}
     \hline
   \end{mytable}
 %\end{table}

- \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}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{tab:08}
+\end{table}

 
 
 \section{Conclusion}
 
 
 \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.
+
+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.
+
+
+In future works, we  plan to investigate how to simulate  the behavior of really
+large scale  applications. For  example, if  we are  interested to  simulate the
+execution of the solvers of this paper with thousand or even dozens of thousands
+or core,  it is not possible  to do that with  SimGrid. In fact, this  tool will
+make the real computation. So we plan to focus our research on that problematic.
+

 
 
 %\section*{Acknowledgment}
 
 
 %\section*{Acknowledgment}


@@ -892,9 +908,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 
 \bibliographystyle{wileyj}
 \bibliography{biblio}
 
 \bibliographystyle{wileyj}
 \bibliography{biblio}

-\AG{Warning bibtex à corriger (%
-  \texttt{empty booktitle in Bru95}%
-).}
+

 
 \end{document}
 
 
 \end{document}