X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/d4ade70c19ff4980df5662db589ac0defced4391..2f78f080350308e2f46d8eff8d66a8e127fee583:/paper.tex

diff --git a/paper.tex b/paper.tex
index c8b8918..f9434f9 100644
--- 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},
@@ -94,27 +94,26 @@ analysis of simulated grid-enabled numerical iterative algorithms}
   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}
 
@@ -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
-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
-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
@@ -165,34 +164,30 @@ 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
-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. 
-
-\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
-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
+parameter variation of the execution platform and of the application data can
+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.
+
+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
+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.
+
+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
@@ -200,22 +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
-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
-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.
 
-\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}
 
-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,
-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}.
 
@@ -225,7 +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
-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
@@ -233,17 +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
-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,
-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
-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
-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.
 
@@ -252,6 +245,57 @@ magnitude. To our knowledge, there is no study on this problematic.
 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}
@@ -273,7 +317,7 @@ where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
 
 \begin{figure}[t]
 %\begin{algorithm}[t]
@@ -342,26 +386,20 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of the two-stage methods using SimGrid toolkit}
 \label{sec:04.02}
 
-One of our objectives when simulating the  application in Simgrid is, as in real
+One of our objectives when simulating the  application in SimGrid is, as in real
 life, to  get accurate results  (solutions of the  problem) but also to ensure the
 test reproducibility  under the same  conditions.  According to  our experience,
-very  few modifications  are required  to adapt  a MPI  program for  the Simgrid
+very  few modifications  are required  to adapt  a MPI  program for  the SimGrid
 simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
-libraries  and related  header files  (smpi.h).  The  second modification  is to
+libraries  and related  header files  (\verb+smpi.h+).  The  second modification  is to
 suppress all global variables by replacing  them with local variables or using a
-Simgrid      selector       called      "runtime       automatic      switching"
+SimGrid selector       called      "runtime       automatic      switching"
 (smpi/privatize\_global\_variables). Indeed, global  variables can generate side
 effects on runtime between the threads running in the same process and generated by
-Simgrid  to simulate the  grid environment.
-
-%\RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
-%last modification on the  MPI program pointed out for some  cases, the review of
-%the sequence of  the MPI\_Isend, MPI\_Irecv and  MPI\_Waitall instructions which
-%might cause an infinite loop.
+SimGrid  to simulate the  grid environment.
 
-
-\paragraph{Simgrid Simulator parameters}
-\  \\ \noindent  Before running  a Simgrid  benchmark, many  parameters for  the
+\paragraph{Parameters of the simulation in SimGrid}
+\  \\ \noindent  Before running  a SimGrid  benchmark, many  parameters for  the
 computation platform must be defined. For our experiments, we consider platforms
 in which  several clusters are  geographically distant,  so there are  intra and
 inter-cluster communications. In the following, these parameters are described:
@@ -369,10 +407,10 @@ inter-cluster communications. In the following, these parameters are described:
 \begin{itemize}
 	\item hostfile: hosts description file.
 	\item platform: file describing the platform architecture: clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}).
 	\item archi   : grid computational description (number of clusters, number of
-nodes/processors for each cluster).
+nodes/processors in each cluster).
 \end{itemize}
 \noindent
 In addition, the following arguments are given to the programs at runtime:
@@ -380,8 +418,8 @@ In addition, the following arguments are given to the programs at runtime:
 \begin{itemize}
 	\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 sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+	\item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
 	\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,
@@ -390,17 +428,18 @@ In addition, the following arguments are given to the programs at runtime:
       	\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.
+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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
-\section{Experimental Results}
+\section{Experimental results}
 \label{sec:expe}
 
-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
 
 \subsection{The 3D Poisson problem}
+\label{3dpoisson}
 
 
 We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form:
@@ -434,9 +473,9 @@ have been chosen for the study in this paper. \\
 
 \textbf{Step 2}: Collect the software materials needed for the experimentation.
 In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting
-method. In addition, the Simgrid simulator has been chosen to simulate the
-behaviors of the distributed applications. Simgrid is running in a virtual
+3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+method. In addition, the SimGrid simulator has been chosen to simulate the
+behaviors of the distributed applications. SimGrid is running in a virtual
 machine on a simple laptop. \\
 
 \textbf{Step 3}: Fix the criteria which will be used for the future
@@ -445,10 +484,10 @@ on the  one hand the algorithm execution mode (synchronous and asynchronous)
 and on the other hand the execution time and the number of iterations to reach the convergence. \\
 
 \textbf{Step 4  }: Set up the  different grid testbed environments  that will be
-simulated in the  simulator tool to run the program.  The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+simulated in the  simulator tool to run the program.  The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
 represents the number  of clusters in the grid and  the second number represents
-the number  of hosts (processors/cores)  in each  cluster. The network  has been
+the number  of hosts (processors/cores)  in each  cluster. The network has been
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
 (resp.  inter-clusters backbone links). \\
@@ -460,8 +499,7 @@ input data.  \\
 
 \textbf{Step 6} : Collect and analyze the output results.
 
-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}
 
 When running a distributed application in a computational grid, many factors may
 have a strong impact on the performance.  First of all, the architecture of the
@@ -474,10 +512,10 @@ 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}
-\item  the network  bandwidth  (bw=bits/s) also  known  as "the  data-carrying
+\item  the network  bandwidth  ($bw$ in bits/s) also  known  as "the  data-carrying
     capacity" of the network is defined as  the maximum of data that can transit
     from one point to another in a unit of time.
-\item the  network latency  (lat :  microsecond) defined as  the delay  from the
+\item the  network latency  ($lat$ in microseconds) defined as  the delay  from the
   start time to send  a simple data from a source to a destination.
 \end{enumerate}
 Upon  the   network  characteristics,  another  impacting   factor  is  the volume of data exchanged  between the nodes in the cluster
@@ -488,8 +526,8 @@ and  between distant  clusters.  This parameter is application dependent.
  on  the other  hand, the  "inter-network" which  is the  backbone link  between
  clusters.  In   practice,  these  two   networks  have  different   speeds.
  The intra-network  generally works  like a  high speed  local network  with a
- high bandwith and very low latency. In opposite, the inter-network connects
- clusters sometime via  heterogeneous networks components  throuth internet with
+ high bandwidth and very low latency. In opposite, the inter-network connects
+ clusters sometime via  heterogeneous networks components  through internet with
  a lower speed.  The network  between distant  clusters might  be a  bottleneck
  for  the global performance of the application.
 
@@ -517,12 +555,12 @@ architectures and scaling up the input matrix size}
 \begin{center}
 \begin{tabular}{r c }
  \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
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
+ Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\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?}
+\caption{Test conditions: various grid configurations with the input matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 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}
@@ -533,31 +571,30 @@ architectures and scaling up the input matrix size}
 
 
 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}
+grid configurations  (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16  and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. First,  the results in  Figure~\ref{fig:01}
 show for all grid configurations the non-variation of the number of iterations of
 classical  GMRES for  a given  input matrix  size; it is not  the case  for the
 multisplitting method.
 
-\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
-\RC{Les légendes ne sont pas explicites...}
+%\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+%\RC{Les légendes ne sont pas explicites...}
 
 
 \begin{figure} [ht!]
   \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}
 
 
-The execution  times between  the two algorithms  is significant  with different
-grid architectures, even  with the same number of processors  (for example, 2x16
-and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+Secondly, the execution  times between  the two algorithms  is significant  with different
+grid architectures, even  with the same number of processors  (for example, 2 $\times$ 16
+and  4 $\times$ 8). We  can  observ  the 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. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+in the  grid: in  average, the reduction of the execution time for GMRES  (resp. Multisplitting)  algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) 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 speeds \\}
 
@@ -565,31 +602,29 @@ $40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC
 \begin{center}
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
+ Inter Networks & 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}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with  networks N1 vs N2}
 \label{tab:02}
 \end{center}
 \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...}
+In this section, the experiments  compare the  behavior of  the algorithms  running on a
+speeder inter-cluster  network (N1) and  also on  a less performant  network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \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
+for  both  algorithms when using  a  grid  architecture  like  4 $\times$ 16 or  8 $\times$ 8: 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}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2
 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
 \label{fig:02}
 \end{figure}
@@ -602,9 +637,9 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\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}
@@ -638,9 +673,9 @@ magnitude with a latency of $8.10^{-6}$.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\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}
@@ -667,9 +702,9 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 4x8\\ %\hline
+ Grid Architecture & 4 $\times$ 8\\ %\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}
@@ -699,7 +734,7 @@ But the interesting results are:
 These  findings may  help a  lot end  users to  setup the  best and  the optimal
 targeted environment for the application deployment when focusing on the problem
 size scale up.  It  should be noticed that the same test has  been done with the
-grid 2x16 leading to the same conclusion.
+grid 2 $\times$ 16 leading to the same conclusion.
 
 \subsubsection{CPU Power impacts on performance}
 
@@ -707,9 +742,9 @@ grid 2x16 leading to the same conclusion.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid architecture & 2x16\\ %\hline
+ Grid architecture & 2 $\times$ 16\\ %\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}
@@ -727,10 +762,16 @@ 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.
+\ \\
+%\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}
@@ -762,11 +803,11 @@ The test conditions are summarized in the table~\ref{tab:07}: \\
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 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
+ 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}
@@ -777,7 +818,7 @@ 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
-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
@@ -792,7 +833,7 @@ geographically distant clusters through the internet.
     \end{tabular}}
 
 
-\begin{figure}[!t]
+\begin{table}[!t]
 \centering
 %\begin{table}
 %  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
@@ -819,14 +860,39 @@ geographically distant clusters through the internet.
     \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}
-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}
@@ -835,10 +901,7 @@ This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-
 
 \bibliographystyle{wileyj}
 \bibliography{biblio}
-\AG{Des warnings bibtex à corriger (%
-  \texttt{entry type for "SimGrid" isn't style-file defined},
-  \texttt{empty booktitle in Bru95}%
-).}
+
 
 \end{document}