DL : encore expe

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index e2dc291f11310a1b0c618c7a025622af21ed388e..31eb0e89348ae7f0c3e7e0b449b81b5aedc7ff2f 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -21,10 +21,11 @@
 \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}

+\usepackage{multirow}
+

 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}


@@ -45,6 +46,8 @@
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RCE}[2][inline]{%
   \todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}
   \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
 \newcommand{\RCE}[2][inline]{%
   \todo[color=yellow!10,#1]{\sffamily\textbf{RCE:} #2}\xspace}

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

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


@@ -70,124 +73,255 @@
 
 
 \begin{document}
 
 
 \begin{document}

-\RCE{Titre a confirmer.}
-\title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+\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
+%% 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.

 
 

-\begin{abstract}
-  The behavior of multicore 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. We have decided to use SimGrid as it enables to benchmark MPI applications.
+%% 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.

 
 

-In this paper, 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 libear  systems. Two different  variantsof the Multisplitting are
-studied: one  using synchronoous  iterations and  another one  with asynchronous
-iterations. For each algorithm we have  tested different parameters to see their
-influence.  We strongly  recommend people  interested  by investing  into a  new
-expensive  hardware  architecture  to   benchmark  their  applications  using  a
-simulation tool before.
+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}
 

-\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
+%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
+%performance}
+\keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}

 
 \maketitle
 
 
 \maketitle
 

-\section{Introduction} 
-The use of multi-core architectures for solving large scientific problems seems to  become imperative  in  a  lot  of  cases. 
-Whatever the scale of these architectures (distributed clusters, computational grids, embedded multi-core \ldots) they  are generally 
-well adapted to execute complexe parallel applications operating on a large amount of data.  Unfortunately,  users (industrials or scientists), 
-who need such computational resources, may not have an easy access to such efficient architectures. The cost of using the platform and/or the cost of 
-testing and deploying an application 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, code debugging, ability to obtain results quickly \ldots at the condition that the simulation results are in education 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 simple. It generally involves the division of the problem
-into  several  \emph{blocks}  that  will  be  solved  in  parallel  on  multiple
-processing units.  Each processing unit has to
-compute an iteration, to send/receive some data dependencies to/from
-its neighbors and to iterate this process until the convergence of
-the method. Several well-known methods demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
-In this processing mode a task cannot begin a new iteration while it
-has not received data dependencies from its neighbors. We say that the iteration computation follows a 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 asynchronous inducing that there is
-no more idle times, due to synchronizations, between two
-iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks that we detail in section 2 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  synchronizations~(see \cite{Bahi07} for more details).
-
-Nevertheless, in both cases (synchronous or asynchronous) 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.~\cite{Calheiros:2011:CTM:1951445.1951450}.   This problematic is even more difficult for the asynchronous scheme 
-where variations of the parameters of the execution platform can lead to very different number of iterations required to 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 solver) 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{ref1} 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  comparing to the synchronous  GMRES. 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 asynchronous iterative application.  
-
-This paper is organized as follows. Section 1 \ref{sec:synchro} 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:04} details the different solvers that we use. 
-Finally our experimental results are presented in section \ref{sec:expe} followed by some concluding remarks and perspectives.
-
-
-\section{The asynchronous iteration model}
+\section{Introduction}  The use of multi-core architectures to solve large
+scientific problems seems to  become imperative  in  many situations.
+Whatever the scale of these architectures (distributed clusters, computational
+grids, embedded multi-core,~\ldots) they  are generally  well adapted to execute
+complex parallel applications operating on a large amount of data.
+Unfortunately,  users (industrials or scientists),  who need such computational
+resources, may not have an easy access to such efficient architectures. The cost
+of using the platform and/or the cost of  testing and deploying an application
+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,
+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
+simple. It generally involves the division of the problem into  several
+\emph{blocks}  that  will  be  solved  in  parallel  on  multiple processing
+units.  Each processing unit has to compute an iteration to send/receive some
+data dependencies to/from its neighbors and to iterate this process until the
+convergence of the method. Several well-known studies demonstrate the
+convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
+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 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
+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
+synchronizations~(see~\cite{bahi07} for more details).
+
+Nevertheless,  in both  cases  (synchronous  or asynchronous)  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~\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 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
+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:04} details the different solvers that we use.  Finally our
+experimental results are presented in Section~\ref{sec:expe} followed by some
+concluding remarks and perspectives.
+
+
+\section{The asynchronous iteration model and the motivations of our work}

 \label{sec:asynchro}
 
 \label{sec:asynchro}
 

+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 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}.
+
+Before  using  an  asynchronous  iterative   method,  the  convergence  must  be
+studied. Otherwise, the  application is not ensure to reach  the convergence. An
+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
+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
+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 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
+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 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.
+

 \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}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Two-stage multisplitting methods}
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}

-In this paper we focus on two-stage multisplitting methods in their both versions synchronous and asynchronous~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+In this paper we focus on two-stage multisplitting methods in their both versions (synchronous and asynchronous)~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$:

 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}
 \begin{equation}
 Ax=b,
 \label{eq:01}
 \end{equation}

-where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). The two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows
+where $A$ is a sparse square and nonsingular matrix, $b$ is the right-hand side and $x$ is the solution of the system. Our work in this paper is restricted to the block Jacobi splitting method. This approach of multisplitting consists in partitioning the matrix $A$ into $L$ horizontal band matrices of order $\frac{n}{L}\times n$ without overlapping (i.e. sub-vectors $\{x_\ell\}_{1\leq\ell\leq L}$ are disjoint). Two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows:

 \begin{equation}
 x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
 \label{eq:02}
 \end{equation}
 \begin{equation}
 x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
 \label{eq:02}
 \end{equation}

-where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system 
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system:

 \begin{equation}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}
 \begin{equation}
 A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \label{eq:03}
 \end{equation}

-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is 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{figure}[htpb]

 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]


@@ -203,29 +337,29 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 \end{algorithmic}
 \caption{Block Jacobi two-stage multisplitting method}
 \label{alg:01}
 \end{algorithmic}
 \caption{Block Jacobi two-stage multisplitting method}
 \label{alg:01}

-%\end{algorithm} 
+%\end{algorithm}

 \end{figure}
 
 \end{figure}
 

-In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on asynchronous model which allows the communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged
+In this paper, we propose two algorithms of two-stage multisplitting methods. The first algorithm is based on the asynchronous model which allows communications to be overlapped by computations and reduces the idle times resulting from the synchronizations. So in the asynchronous mode, our two-stage algorithm uses asynchronous outer iterations and asynchronous communications between clusters. The communications (i.e. lines~\ref{send} and~\ref{recv} in Figure~\ref{alg:01}) are performed by message passing using MPI non-blocking communication routines. The convergence of the asynchronous iterations is detected when all clusters have locally converged:

 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}
 \begin{equation}
 k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
 \label{eq:04}

-\end{equation}    
-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 
+\end{equation}
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.

 
 

-The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration 
+The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration:

 \begin{equation}
 S=[x^1,x^2,\ldots,x^s],~s\ll n.
 \label{eq:05}
 \begin{equation}
 S=[x^1,x^2,\ldots,x^s],~s\ll n.
 \label{eq:05}

-\end{equation}   
-At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+\end{equation}
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual:

 \begin{equation}
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}
 \begin{equation}
 \min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
 \label{eq:06}

-\end{equation} 
+\end{equation}

 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).

- 
-\begin{figure}[t]
+
+\begin{figure}[htbp]

 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]


@@ -240,7 +374,7 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
        \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
        \State $\tilde{x_\ell}=S_\ell\alpha$
        \State Send $\tilde{x_\ell}$ to neighboring clusters
        \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
        \State $\tilde{x_\ell}=S_\ell\alpha$
        \State Send $\tilde{x_\ell}$ to neighboring clusters

-       \Else 
+       \Else

          \State Send $x_\ell^k$ to neighboring clusters
     \EndIf
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
          \State Send $x_\ell^k$ to neighboring clusters
     \EndIf
     \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters


@@ -248,396 +382,353 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \end{algorithmic}
 \caption{Krylov two-stage method using block Jacobi multisplitting}
 \label{alg:02}
 \end{algorithmic}
 \caption{Krylov two-stage method using block Jacobi multisplitting}
 \label{alg:02}

-%\end{algorithm} 
+%\end{algorithm}

 \end{figure}
 
 \end{figure}
 

-\subsection{Simulation of two-stage methods using SimGrid framework}
+\subsection{Simulation of the two-stage methods using SimGrid toolkit}

 \label{sec:04.02}
 
 \label{sec:04.02}
 

-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 ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in SIMGRID simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a 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, generated by the Simgrid to simulate the grid environment.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.     
-
-
-\paragraph{SIMGRID Simulator parameters}
+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
+simulator using SMPI (Simulator MPI). The  first modification is to include SMPI
+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"
+(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.
+
+\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:

 
 \begin{itemize}
 
 \begin{itemize}

-       \item hostfile: Hosts description file.
-       \item plarform: File describing the platform architecture : clusters (CPU power,
-\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). 
+       \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{}),
+       \item archi   : grid computational description (number of clusters, number of
+nodes/processors in each cluster).

 \end{itemize}
 \end{itemize}

-
-
+\noindent

 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 Matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
-       \item Matrix diagonal value = 6.0;
-       \item Execution Mode: synchronous or asynchronous.
+       \item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
+       \item inner precision $\TOLG$ and outer precision $\TOLM$,
+       \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,
+        \item maximum number of iterations and precision for the classical GMRES method,
+        \item maximum number of restarts for the Arnorldi process in GMRES method,
+       \item execution mode: synchronous or asynchronous.

 \end{itemize}
 
 \end{itemize}
 

-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determine 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}
 
 \label{sec:expe}
 

+In this section, experiments for both multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.

 
 

-\subsection{Setup study and Methodology}
+\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:
+\begin{equation}
+\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
+\label{eq:07}
+\end{equation}
+such that:
+\begin{equation*}
+\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
+\end{equation*}
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that:
+\begin{equation}
+\begin{array}{ll}
+\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
+\end{array}
+\label{eq:08}
+\end{equation}
+until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.

 
 

-To conduct our study, we have put in place the following methodology 
-which can be reused for any grid-enabled applications.
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
+sub-problems such that $L$ is the number of clusters and $p$ is the number of
+processors in each cluster. We apply the three-dimensional partitioning instead
+of the row-by-row one in order to reduce the size of the data shared at the
+sub-problems boundaries. In this case, each processor is in charge of
+parallelepipedic block of the problem and has at most six neighbors in the same
+cluster or in distant clusters with which it shares data at boundaries.

 
 

-\textbf{Step 1} : Choose with the end users the class of algorithms or 
-the application to be tested. Numerical parallel iterative algorithms 
+\subsection{Study setup and simulation methodology}
+
+First, to conduct our study, we propose the following methodology
+which can be reused for any grid-enabled applications.\\
+
+\textbf{Step 1}: Choose with the end users the class of algorithms or
+the application to be tested. Numerical parallel iterative algorithms

 have been chosen for the study in this paper. \\
 
 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 three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, SIMGRID simulator has been chosen to simulate the behaviors of the 
-distributed applications. SIMGRID is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
-
-\textbf{Step 3} : Fix the criteria which will be used for the future 
-results comparison and analysis. In the scope of this study, we retain 
-in one hand the algorithm execution mode (synchronous and asynchronous) 
-and in the other hand the execution time and the number of iterations of 
-the application before obtaining the convergence. \\
-
-\textbf{Step 4 }: Setup up the different grid testbeds environment 
-which will be simulated in the simulator tool to run the program. The 
-following architecture has been configured in Simgrid : 2x16 - that is a 
-grid containing 2 clusters with 16 hosts (processors/cores) each -, 4x8, 
-4x16, 8x8 and 2x50. The network has been designed to operate with a 
-bandwidth equals to 10Gbits (resp. 1Gbits/s) and a latency of 8E-6 
-microseconds (resp. 5E-5) for the intra-clusters links (resp. 
-inter-clusters backbone links). \\
-
-\textbf{Step 5}: Conduct an extensive and comprehensive testings 
-within these configurations in varying the key parameters, especially 
-the CPU power capacity, the network parameters and also the size of the 
-input matrix. Note that some parameters should be fixed to be invariant to allow the 
-comparison like some program input arguments. \\
+\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
+machine on a simple laptop. \\
+
+\textbf{Step 3}: Fix the criteria which will be used for the future
+results comparison and analysis. In the scope of this study, we retain
+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 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. \\
+
+\textbf{Step 5}: Conduct an extensive and comprehensive testings
+within these configurations by varying the key parameters, especially
+the CPU power capacity, the network parameters and also the size of the
+input data.  \\

 
 \textbf{Step 6} : Collect and analyze the output results.
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 

-\subsection{Factors impacting distributed applications performance in 
-a grid environment}
-
-From our previous experience on running distributed application in a 
-computational grid, many factors are identified to have an impact on the 
-program behavior and performance on this specific environment. Mainly, 
-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. 
-
-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 : (i) the network 
-bandwidth (bw=bits/s) also known as "the data-carrying capacity" 
-of the network is defined as the maximum of data that can pass 
-from one point to another in a unit of time. (ii) the network latency 
-(lat : microsecond) defined as the delay from the start time to send the 
-data from a source and the final time the destination have finished to 
-receive it. Upon the network characteristics, another impacting factor 
-is the application dependent volume of data exchanged between the nodes 
-in the cluster and between distant clusters. Large volume of data can be 
-transferred in transit between the clusters and nodes during the code 
-execution. 
-
- In a grid environment, it is common to distinguish in one hand, the 
-"\,intra-network" which refers to the links between nodes within a 
-cluster and in the other hand, the "\,inter-network" which is the 
-backbone link between clusters. By design, these two networks perform 
-with different speed. 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 thru internet with a lower speed. The network 
-between distant clusters might be a bottleneck for the global 
-performance of the application. 
-
-\subsection{Comparing GMRES and Multisplitting algorithms in 
+\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
+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 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  ($bw$ 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  ($lat$ 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
+and  between distant  clusters.  This parameter is application dependent.
+
+ In  a grid  environment, it  is common  to distinguish,  on one hand,  the
+ \textit{intra-network} which refers  to the links between nodes within  a
+ cluster and on  the other  hand, the  \textit{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 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.
+
+
+\subsection{Comparison between GMRES and two-stage multisplitting algorithms in

 synchronous mode}
 synchronous mode}

+In the scope of this paper, our first objective is to analyze
+when the synchronous Krylov two-stage 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.
+
+Table~\ref{tab:01} summarizes the parameters used in the different simulations:
+the grid architectures (i.e. the number of clusters and the number of nodes per
+cluster), the network of inter-clusters backbone links and the matrix sizes of
+the 3D Poisson problem. However, for all simulations we fix the network
+parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
+$lat=8\mu$s. In what follows, we will present the test conditions, the output
+results and our comments.
+
+\begin{table} [ht!]
+\begin{center}
+\begin{tabular}{ll}
+\hline
+Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{Matrix size}            & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
+                                        & $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
+\end{tabular}
+\caption{Parameters for the different simulations}
+\label{tab:01}
+\end{center}
+\end{table}

 
 

-In the scope of this paper, our first objective is to demonstrate the 
-Algo-2 (Multisplitting method) shows a better performance in grid 
-architecture compared with Algo-1 (Classical GMRES) both running in 
-\textbf{\textit{synchronous mode}}. Better algorithm performance 
-should means a less number of iterations output and a less execution time 
-before reaching the convergence. For a systematic study, the experiments 
-should figure out that, for various grid parameters values, the 
-simulator will confirm the targeted outcomes, particularly for poor and 
-slow networks, focusing on the impact on the communication performance 
-on the chosen class of algorithm.
-
-The following paragraphs present the test conditions, the output results 
-and our comments.\\
-
-
-\textit{3.a Executing the algorithms on various computational grid 
-architecture scaling up the input matrix size}
-\\
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 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}
-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
-
-\end{footnotesize}
-
-
- 
-%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-
-
-The results in figure 3 show 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. 
-
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
-\centering
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
+
+In  this  section,  we  analyze   the  simulations  conducted  on  various  grid
+configurations and for different sizes of the 3D Poisson problem. The parameters
+of    the    network    between    clusters    is    fixed    to    $N2$    (see
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
+a given matrix size of 170$^3$ elements, a  non-variation in the number of
+iterations for the classical GMRES algorithm, which is not the case of the
+Krylov two-stage algorithm. In fact, with multisplitting  algorithms, the number
+of splitting (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number  of splitting is, the slower the
+convergence of the algorithm is (see the output results obtained from
+configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
+8$\times$8).
+
+The execution times between both algorithms is significant with different grid
+architectures. The synchronous Krylov two-stage algorithm presents better
+performances than the GMRES algorithm, even for a high number of clusters (about
+$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+observe a better sensitivity of the Krylov two-stage algorithm (compared to the
+GMRES one) when scaling up the number of the processors in the computational
+grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
+about $40\%$ better on $64$ processors (grid of 8$\times$8) than $32$ processors
+(grid of 2$\times$16).
+
+\begin{figure}[ht]
+\begin{center}

 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}

-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170} 
-%\label{overflow}}
+\end{center}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
+\label{fig:01}

 \end{figure}
 \end{figure}

-%\end{wrapfigure}
-
-Unless the 8x8 cluster, the time 
-execution difference between the two algorithms is important when 
-comparing between different grid architectures, even with the same number of 
-processors (like 2x16 and 4x8 = 32 processors for example). The 
-experiment concludes the low sensitivity of the multisplitting method 
-(compared with the classical GMRES) when scaling up to higher input 
-matrix size. 
-
-\textit{\\3.b Running on various computational grid architecture\\}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 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}
-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
-

 
 

-
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\subsubsection{Simulations for two different inter-clusters network speeds\\}
+In  Figure~\ref{fig:02} we  present the  execution times  of both  algorithms to
+solve a  3D Poisson problem of  size $150^3$ on two  different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure  that the Krylov two-stage  algorithm is sensitive to  the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an  interesting behavior of  the Krylov  two-stage algorithm. It  is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the  GMRES algorithms.  This  means  that the  multisplitting  methods are  more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

-\caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+\RCE{ok}
+\label{fig:02}

 \end{figure}
 \end{figure}

-%\end{wrapfigure}
-
-The experiments compare the behavior of the algorithms running first on 
-a speed inter- cluster network (N1) and a less performant network (N2). 
-Figure 4 shows that end users will gain to reduce the execution time 
-for both algorithms in using a grid architecture like 4x16 or 8x8: the 
-performance was increased in a factor of 2. The results depict also that 
-when the network speed drops down, the difference between the execution 
-times can reach more than 25\%. 
-
-\textit{\\3.c Network latency impacts on performance\\}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 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}
-Table 3 : Network latency impact \\
-
-\end{footnotesize}

 
 

+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.

 
 

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

 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}

-\caption{Network latency impact on execution time}
-%\label{overflow}}
+\caption{Network latency impacts on performances}
+\label{fig:03}

 \end{figure}
 
 \end{figure}
 

+\subsubsection{Network bandwidth impacts on performances\\}

 
 

-According the results in figure 5, degradation of the network 
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time 
-increase more than 75\% (resp. 82\%) of the execution for the classical 
-GMRES (resp. multisplitting) algorithm. In addition, it appears that the 
-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 of the time for 
-the multisplitting, even though, the performance was on the same order 
-of magnitude with a latency of 8.10$^{-6}$. 
-
-\textit{\\3.d Network bandwidth impacts on performance\\}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 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}
-Table 4 : Network bandwidth impact \\
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
+solve a 3D Poisson problem of size $150^3$. The results of increasing the
+network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain in the considered bandwidth
+interval with a gain of $40\%$ compared to only about $24\%$ for the classical
+GMRES algorithm.

 
 

-\end{footnotesize}
-
-
-\begin{figure} [ht!]
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}

-\caption{Network bandwith impact on execution time}
-%\label{overflow}
+\caption{Network bandwith impacts on performances}
+\label{fig:04}

 \end{figure}
 
 \end{figure}
 

-
-
-The results of increasing the network bandwidth depict the improvement 
-of the performance by reducing the execution time for both of the two 
-algorithms (Figure 6). However, and again in this case, the multisplitting method 
-presents a better performance in the considered bandwidth interval with 
-a gain of 40\% which is only around 24\% for classical GMRES.
-
-\textit{\\3.e Input matrix size impacts on performance\\}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
- \end{tabular}
-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
-
-
-\begin{figure} [ht!]
+\subsubsection{Matrix size impacts on performances\\}
+
+In these experiments, the matrix size of the 3D Poisson problem is varied from
+$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
+clusters of $8$ processors each interconnected by the network $N2$ (see
+Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes.  For all problem
+sizes, the GMRES algorithm is always slower than the Krylov two-stage algorithm.
+Moreover, for this benchmark, it seems that the greater the problem size is, the
+bigger the ratio between execution times of both algorithms is. We can also
+observe that for some problem sizes, the convergence (and thus the execution
+time) of the Krylov two-stage algorithm varies quite a lot.
+%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+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.
+
+\begin{figure}[ht]

 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}

-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impacts on performances}
+\label{fig:05}

 \end{figure}
 
 \end{figure}
 

-In this experimentation, the input matrix size has been set from 
-N$_{x}$ = N$_{y}$ = N$_{z}$ = 40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
-200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 7, 
-the execution time for the two algorithms convergence increases with the 
-input matrix size. But the interesting results here direct on (i) the 
-drastic increase (300 times) of the number of iterations needed before 
-the convergence for the classical GMRES algorithm when the matrix size 
-go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost 
-the double from N$_{x}$=140 compared with the convergence time of the 
-multisplitting method. 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. Note that the 
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{\\3.f CPU Power impact on performance\\}
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
- \end{tabular}
-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
+\subsubsection{CPU power impacts on performances\\}
+Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.

 
 

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

 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}

-\caption{CPU Power impact on execution time}
-%\label{overflow}}
+\caption{CPU Power impacts on performances}
+\label{fig:06}

 \end{figure}
 \end{figure}

-
-Using the SIMGRID simulator flexibility, we have tried to determine the 
-impact on the algorithms performance in varying the CPU power of the 
-clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6 
-confirm the performance gain, around 95\% for both of the two methods, 
-after adding more powerful CPU. Note that the execution time axis in the 
-figure is in logarithmic scale.
-
-\subsection{Comparing GMRES in native synchronous mode and 
-Multisplitting algorithms in asynchronous mode}
-
-The previous paragraphs put in evidence the interests to simulate the 
-behavior of the application before any deployment in a real environment. 
-We have focused the study on analyzing the performance in varying the 
-key factors impacting the results. In the same line, the study compares 
-the performance of the two proposed methods in \textbf{synchronous mode
-}. In this section, with the same previous methodology, the goal is to 
-demonstrate the efficiency of the multisplitting method in \textbf{
-asynchronous mode} compare with the classical GMRES staying in the 
-synchronous mode.
-
-Note that the interest of using the asynchronous mode for data exchange 
-is mainly, in opposite of the synchronous mode, the non-wait aspects of 
-the current computation after a communication operation like sending 
-some data between nodes. Each processor can continue their local 
-calculation without waiting for the end of the communication. Thus, the 
-asynchronous may theoretically reduce the overall execution time and can 
-improve the algorithm performance.
-
-As stated supra, SIMGRID simulator tool has been used to prove the 
-efficiency of the multisplitting in asynchronous mode and to find the 
-best combination of the grid resources (CPU, Network, input matrix size, 
-\ldots ) to get the highest "\,relative gain" in comparison with the 
-classical GMRES time. 
-
-
-The test conditions are summarized in the table below : \\
-
-% environment
-\begin{footnotesize}
-\begin{tabular}{r c }
- \hline  
- Grid & 2x50 totaling 100 processors\\ %\hline
- Processors & 1 GFlops to 1.5 GFlops\\
-   Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
-   Inter-Network & bw=5 Mbits - lat=2E-02\\
- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+\ \\
+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 two-stage algorithms is in accordance  with larger real
+executions on large scale supercomputers~\cite{couturier15}.
+
+
+\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
+
+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} compared with the
+classical GMRES in \textit{synchronous mode}.
+
+The  interest of  using  an asynchronous  algorithm  is that  there  is no  more
+synchronization. With  geographically distant  clusters, this may  be essential.
+In  this case,  each  processor can  compute its  iteration  freely without  any
+synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
+theoretically reduce  the overall execution  time and can improve  the algorithm
+performance.
+
+In this section,  the SimGrid simulator is  used to compare the  behavior of the
+two-stage algorithm in  asynchronous mode  with GMRES  in synchronous  mode.  Several
+benchmarks have  been performed with  various combinations of the  grid resources
+(CPU, Network, matrix size, \ldots). The test  conditions are summarized
+in  Table~\ref{tab:02}. In  order to  compare  the execution  times, Table~\ref{tab:03}
+reports the  relative gain between both  algorithms. It is defined  by the ratio
+between  the   execution  time  of   GMRES  and   the  execution  time   of  the
+multisplitting.
+\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+\RCE{Table III avec la nouvelle numerotation}
+The  ratio  is  greater  than  one  because  the  asynchronous
+multisplitting version is faster than GMRES.
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{ll}
+ \hline
+ Grid architecture                       & 2$\times$50 totaling 100 processors\\
+ Processors Power                        & 1 GFlops to 1.5 GFlops \\
+ \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
+                                         & $bw$=5 Mbits, $lat=20ms$s\\
+ Matrix size                             & from $62^3$ to $150^3$\\
+ Residual error precision                & $10^{-5}$ to $10^{-9}$\\ \hline \\

  \end{tabular}
  \end{tabular}

-\end{footnotesize}
-
-Again, comprehensive and extensive tests have been conducted varying the 
-CPU power and the network parameters (bandwidth and latency) in the 
-simulator tool with different problem size. The relative gains greater 
-than 1 between the two algorithms have been captured after each step of 
-the test. Table I below has recorded the best grid configurations 
-allowing a multiplitting method time more than 2.5 times lower than 
-classical GMRES execution and convergence time. The finding thru this 
-experimentation is the tolerance of the multisplitting method under a 
-low speed network that we encounter usually with distant clusters thru the 
-internet.
+\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\label{tab:02}
+\end{table}
+

 
 % 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}}


@@ -647,72 +738,86 @@ internet.
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 

-\begin{table}[!t]
-  \centering
-  \caption{Relative gain of the multisplitting algorithm compared with 
-the classical GMRES}
-  \label{"Table 7"}

 
 

-  \begin{mytable}{6}
-    \hline
-    bandwidth (Mbit/s)
-    & 5         & 5         & 5         & 5         & 5 \\
-    \hline
-    latency (ms)
-    & 20      & 20      & 20      & 20      & 20 \\
-    \hline
-    power (GFlops)
-    & 1         & 1         & 1         & 1.5       & 1.5 \\
-    \hline
-    size (N)
-    & 62        & 62        & 62        & 100       & 100 \\
-    \hline
-    Precision
-    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\
-    \hline
-    Relative gain
-    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 \\
-    \hline
-  \end{mytable}
-
-  \smallskip
-
-  \begin{mytable}{6}
+\begin{table}[!t]
+\centering
+%\begin{table}
+%  \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+%  \label{"Table 7"}
+ \begin{mytable}{11}

     \hline
     bandwidth (Mbit/s)
     \hline
     bandwidth (Mbit/s)

-    & 50        & 50        & 50        & 50        & 50 \\
+    & 5     & 5     & 5         & 5         & 5  & 50        & 50        & 50        & 50        & 50 \\

     \hline
     latency (ms)
     \hline
     latency (ms)

-    & 20      & 20      & 20      & 20      & 20 \\
+    & 20      & 20      & 20      & 20      & 20 & 20      & 20      & 20      & 20      & 20 \\

     \hline
     power (GFlops)
     \hline
     power (GFlops)

-    & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
+    & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\

     \hline
     \hline

-    size (N)
-    & 110       & 120       & 130       & 140       & 150 \\
+    size ($N^3$)
+    & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\

     \hline
     Precision
     \hline
     Precision

-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
+    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\

     \hline
     Relative gain
     \hline
     Relative gain

-    & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\
+    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\

     \hline
   \end{mytable}
     \hline
   \end{mytable}

+%\end{table}
+ \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \label{tab:03}

 \end{table}
 
 \end{table}
 

-\section{Conclusion}
-CONCLUSION
-
+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
+Table~\ref{tab:08}  are  reported the  best  grid  configurations allowing
+the  two-stage multisplitting algorithm to  be more than  $2.5$ times faster  than the
+classical  GMRES.  These  experiments also  show the  relative tolerance  of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters through the internet.

 
 

-\section*{Acknowledgment}
-
-
-The authors would like to thank\dots{}

 
 

+\section{Conclusion}
+In this paper we have presented the simulation of the execution of three
+different parallel solvers on some multi-core architectures. We have shown 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
+of cores,  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}
+\ack
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

 
 \bibliographystyle{wileyj}
 \bibliography{biblio}
 
 
 \bibliographystyle{wileyj}
 \bibliography{biblio}
 

+

 \end{document}
 
 %%% Local Variables:
 \end{document}
 
 %%% Local Variables: