Desription du problème à résoudre: 3D Poisson equation

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -69,9 +69,8 @@
 
 
 
 
 
 

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

 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{    Charles Emile Ramamonjisoa and
 %\itshape{\journalnamelc}\footnotemark[2]}
 
 \author{    Charles Emile Ramamonjisoa and


@@ -82,7 +81,7 @@
 }
 
 \address{
 }
 
 \address{

-       \centering    
+       \centering

     Femto-ST Institute - DISC Department\\
     Université de Franche-Comté\\
     Belfort\\
     Femto-ST Institute - DISC Department\\
     Université de Franche-Comté\\
     Belfort\\


@@ -91,32 +90,131 @@
 
 %% Lilia Ziane Khodja: Department of Aerospace \& Mechanical Engineering\\ Non Linear Computational Mechanics\\ University of Liege\\ Liege, Belgium. 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 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.
+\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. We have decided to use SimGrid as it enables to benchmark MPI
+applications.

 
 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.
 
 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.
+These algorithms  are used to  solve libear  systems. Two different  variants of
+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.
+
+

 
 

-  

 
 

-  

 \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} 
+\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,~\ldots. 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 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 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 can lead to very different numbers
+of iterations to reach the converge and so to very different execution times. In
+this challenging context we think that the  use of a simulation tool can greatly
+leverage the possibility of testing various platform scenarios.
+
+The main contribution of this paper is to show that the use of a simulation tool
+(i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real  parallel
+applications (i.e. large linear system solvers) can help developers to better
+tune their application for a given multi-core architecture. To show the validity
+of this approach we first compare the simulated execution of the multisplitting
+algorithm  with  the  GMRES   (Generalized   Minimal  Residual)
+solver~\cite{saad86} in synchronous mode. The obtained results on different
+simulated multi-core architectures confirm the real results previously obtained
+on non simulated architectures.  We also confirm  the efficiency  of the
+asynchronous  multisplitting algorithm  compared to the synchronous  GMRES. In
+this way and with a simple computing architecture (a laptop) SimGrid allows us
+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}
 
 \section{The asynchronous iteration model}

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

 
 \section{SimGrid}
 
 \section{SimGrid}

+ \label{sec:simgrid}

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


@@ -125,22 +223,22 @@ simulation tool before.
 \label{sec:04}
 \subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
 \label{sec:04.01}
 \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 ({\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}.

 
 \begin{figure}[t]
 %\begin{algorithm}[t]
 
 \begin{figure}[t]
 %\begin{algorithm}[t]


@@ -158,28 +256,28 @@ 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 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

 \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}    
+\end{equation}

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

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


@@ -195,7 +293,7 @@ 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


@@ -203,189 +301,225 @@ 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}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 
 \end{figure}
 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \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  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  (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, generated by
+the Simgrid  to simulate the  grid environment.  \RC{On vire cette  phrase ?}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}
+\  \\ \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,
+       \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). 
+       \item archi   : grid computational description (number of clusters, number of
+nodes/processors for 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 and outer iterations;
+       \item inner and outer precisions;
+       \item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$);
+       \item matrix diagonal value = 6.0 (for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments); \RC{CE tu vérifies, je dis ca de tête}
+       \item 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 -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 problem sued in our experiments is described.
+
+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+h,y,z) \\
+                  & +\phi(x,y-h,z)+\phi(x,y+h,z) \\
+                  & +\phi(x,y,z-h)+\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. 

 
 

+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 sub-problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries. 

 
 

-\subsection{Setup study and Methodology}
+\subsection{Study setup and Simulation Methodology}

 
 

-To conduct our study, we have put in place the following methodology 
-which can be reused for any grid-enabled applications.
+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 
+\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 on the Mesocentre datacenter in the University of  Franche-Comte and 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
+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
+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
+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). \\
+
+\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 
+\subsection{Factors impacting distributed applications performance in

 a grid environment}
 
 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 
+When running a distributed application in a computational grid, many factors may
+have a strong impact on the performances.  First of all, the architecture of the
+grid itself can obviously influence the  performance results of the program. The
+performance gain  might be important  theoretically when the number  of clusters
+and/or  the  number  of  nodes (processors/cores)  in  each  individual  cluster
+increase.
+
+Another important factor  impacting the overall performances  of the application
+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
+    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
+  start time to send  the data from a source and the  final time the destination
+  have finished to receive it.
+\end{enumerate}
+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  and
+transit between the clusters and nodes during the code execution.
+
+ In  a grid  environment, it  is common  to distinguish,  on the  one hand,  the
+ "intra-network" which refers  to the links between nodes within  a cluster and,
+ on  the other  hand, the  "inter-network" which  is the  backbone link  between
+ clusters.  In   practse;  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  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

 synchronous mode}
 
 synchronous mode}
 

-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 mean 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 
+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
+\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.
 
 on the chosen class of algorithm.
 

-The following paragraphs present the test conditions, the output results 
-and our comments.
+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}
+\textit{3.a Executing the algorithms on various computational grid
+architecture and scaling up the input matrix size}

 \\
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 \\
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline

- Network & N2 : bw=1Gbs-lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
- - & N$_{x}$ =170 x 170 x 170    \\ \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}
  \end{tabular}

-Table 1 : Clusters x Nodes with NX=150 or NX=170 \\
+Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\

 
 \end{footnotesize}
 
 
 
 \end{footnotesize}
 
 

- 
+

 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 
 %\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
 
 

-The results in figure 1 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. 
+In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration 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
 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
 
 %\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}

-\caption{Cluster x Nodes NX=150 and NX=170} 
+\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}

 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}
 
 %\label{overflow}}
 \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. 
+The execution time 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 the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs 40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.

 
 

-\textit{\\3.b Running on various computational grid architecture\\}
+\textit{\\3.b Running on two different speed cluster inter-networks\\}

 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16, 4x8\\ %\hline
  Grid & 2x16, 4x8\\ %\hline

- Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
- - & N2 : bw=1Gbs-lat=5E-05 \\
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \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{tabular}
 Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 


@@ -402,25 +536,24 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-The experiments compare the behavior of the algorithms running first on 
-speed inter- cluster network (N1) and a less performant network (N2). 
-The figure 2 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\%. 
+The experiments compare the behavior of the algorithms running first on
+a speed inter- cluster network (N1) and also on a less performant network (N2).
+Figure 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 (12.5\%), 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 }
 
 \textit{\\3.c Network latency impacts on performance\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline

- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline\\

  \end{tabular}
  \end{tabular}

-

 Table 3 : Network latency impact \\
 
 \end{footnotesize}
 Table 3 : Network latency impact \\
 
 \end{footnotesize}


@@ -435,27 +568,26 @@ Table 3 : Network latency impact \\
 \end{figure}
 
 
 \end{figure}
 
 

-According the results in table and figure 3, 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. 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$^{-6}$. 
+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 }
 
 \textit{\\3.d Network bandwidth impacts on performance\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Grid & 2x16\\ %\hline

- Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \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}
  \end{tabular}

-

 Table 4 : Network bandwidth impact \\
 
 \end{footnotesize}
 Table 4 : Network bandwidth impact \\
 
 \end{footnotesize}


@@ -470,21 +602,18 @@ Table 4 : Network bandwidth impact \\
 
 
 
 
 
 

-The results of increasing the network bandwidth depict the improvement 
-of the performance by reducing the execution time for both of the two 
-algorithms. 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.
+The results of increasing the network bandwidth show the improvement
+of the performance for both of the two algorithms by reducing the execution time (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 }
 
 \textit{\\3.e Input matrix size impacts on performance\\}
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 4x8\\ %\hline
  Grid & 4x8\\ %\hline

- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \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{tabular}
 Table 5 : Input matrix size impact\\
 


@@ -498,18 +627,18 @@ Table 5 : Input matrix size impact\\
 %\label{overflow}}
 \end{figure}
 
 %\label{overflow}}
 \end{figure}
 

-In this experimentation, the input matrix size has been set from 
-Nx=Ny=Nz=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 5, 
-the execution time for the algorithms convergence increases with the 
-input matrix size. But the interesting result 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 Nx=150; (ii) the classical GMRES execution time also almost 
-the double from Nx=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 
+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
+iinput 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\\}
 same test has been done with the grid 2x16 getting the same conclusion.
 
 \textit{\\3.f CPU Power impact on performance\\}


@@ -517,9 +646,9 @@ same test has been done with the grid 2x16 getting the same conclusion.
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x16\\ %\hline
  Grid & 2x16\\ %\hline

- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline

  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 Table 6 : CPU Power impact \\
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
 Table 6 : CPU Power impact \\


@@ -532,41 +661,39 @@ Table 6 : CPU Power impact \\
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 %\label{overflow}}
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
 %\label{overflow}}

-\end{figure}
+s\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.
+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. 

 
 

-\subsection{Comparing GMRES in native synchronous mode and 
+\subsection{Comparing GMRES in native synchronous mode and

 Multisplitting algorithms in asynchronous mode}
 
 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 
+The previous paragraphs put in evidence the interests to simulate the
+behavior of the application before any deployment in a real environment.
+We have focused the study on analyzing the performance in varying the
+key factors impacting the results. The study compares
+the performance of the two proposed algorithms both in \textit{synchronous mode
+}. In this section, following the same previous methodology, the goal is to
+demonstrate the efficiency of the multisplitting method in \textit{
+asynchronous mode} compared with the classical GMRES staying in
+\textit{synchronous mode}.
+
+Note that the interest of using the asynchronous mode for data exchange
+is mainly, in opposite of the synchronous mode, the non-wait aspects of
+the current computation after a communication operation like sending
+some data between nodes. Each processor can continue their local
+calculation without waiting for the end of the communication. Thus, the
+asynchronous may theoretically reduce the overall execution time and can

 improve the algorithm performance.
 
 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. 
+As stated supra, Simgrid simulator tool has been used to prove the
+efficiency of the multisplitting in asynchronous mode and to find the
+best combination of the grid resources (CPU, Network, input matrix size,
+\ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.

 
 
 The test conditions are summarized in the table below : \\
 
 
 The test conditions are summarized in the table below : \\


@@ -574,26 +701,23 @@ The test conditions are summarized in the table below : \\
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }

- \hline  
+ \hline

  Grid & 2x50 totaling 100 processors\\ %\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\\
+ 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 \\
+ Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\

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

 
 % 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}}


@@ -603,55 +727,33 @@ internet.
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 
                   |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
     \end{tabular}}
 

+

 \begin{table}[!t]
   \centering
 \begin{table}[!t]
   \centering

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

 
 

-  \begin{mytable}{6}
+  \begin{mytable}{11}

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

-    & 5         & 5         & 5         & 5         & 5 \\
+    & 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         & 1         & 1         & 1.5       & 1.5 \\
+    & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\

     \hline
     size (N)
     \hline
     size (N)

-    & 62        & 62        & 62        & 100       & 100 \\
+    & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\

     \hline
     Precision
     \hline
     Precision

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

-    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 \\
-    \hline
-  \end{mytable}
-
-  \smallskip
-
-  \begin{mytable}{6}
-    \hline
-    bandwidth (Mbit/s)
-    & 50        & 50        & 50        & 50        & 50 \\
-    \hline
-    latency (ms)
-    & 20      & 20      & 20      & 20      & 20 \\
-    \hline
-    power (GFlops)
-    & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
-    \hline
-    size (N)
-    & 110       & 120       & 130       & 140       & 150 \\
-    \hline
-    Precision
-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
-    \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}
 \end{table}
     \hline
   \end{mytable}
 \end{table}


@@ -662,8 +764,7 @@ CONCLUSION
 
 \section*{Acknowledgment}
 
 
 \section*{Acknowledgment}
 

-
-The authors would like to thank\dots{}
+This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).

 
 
 \bibliographystyle{wileyj}
 
 
 \bibliographystyle{wileyj}