X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/4f3f422c55e11ca375bf5f17203c5d82ed9928aa..46eda615134cd2d70c43d72f1e22ade13bc797fa:/paper.tex

diff --git a/paper.tex b/paper.tex
index 377a87f..8fd9fb4 100644
--- a/paper.tex
+++ b/paper.tex
@@ -90,7 +90,7 @@ analysis of simulated grid-enabled numerical iterative algorithms}
 
 %% 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
+\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
@@ -101,7 +101,7 @@ 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.
-These algorithms  are used to  solve libear  systems. Two different  variantsof
+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
@@ -114,13 +114,13 @@ their  applications  using  a simulation tool before.
 \end{abstract}
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
-%performance} 
-\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}
+%performance}
+\keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}
 
 \maketitle
 
-\section{Introduction}  The use of multi-core architectures for solving large
-scientific problems seems to  become imperative  in  a  lot  of  cases.
+\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.
@@ -131,39 +131,38 @@ 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.
+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
+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 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 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~\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 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
+from its neighbors. We say that the iteration computation follows a
+\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
+iteration without having to wait for the data dependencies coming from its
+neighbors. Both communication and computations are \textit{asynchronous}
+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
@@ -175,7 +174,7 @@ 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  comparing to the synchronous  GMRES. In
+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
@@ -193,6 +192,27 @@ concluding remarks and perspectives.
 \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}
  \label{sec:simgrid}
 
@@ -203,22 +223,22 @@ concluding remarks and perspectives.
 \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}
-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}
-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}
-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, has been studied by many authors for example~\cite{Bru95,bahi07}.
 
 \begin{figure}[t]
 %\begin{algorithm}[t]
@@ -239,19 +259,19 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\end{algorithm}
 \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}
 \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}
 \end{equation}
-At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+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}
@@ -284,35 +304,60 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 %\end{algorithm}
 \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}
 
-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.
+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  (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.
+
+%\RC{On vire cette  phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}The
+%last modification on the  MPI program pointed out for some  cases, the review of
+%the sequence of  the MPI\_Isend, MPI\_Irecv and  MPI\_Waitall instructions which
+%might cause an infinite loop.
 
 
 \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}
-	\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{}).
-	\item archi   : Grid computational description (Number of clusters, Number of
+	\item archi   : grid computational description (number of clusters, number of
 nodes/processors for each cluster).
 \end{itemize}
-
-
+\noindent
 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 maximum number of the GMRES restarts in the Arnorldi process;
+	\item maximum number of iterations and the tolerance threshold in classical GMRES;
+	\item tolerance threshold for outer and inner-iterations;
+	\item matrix size (N$_{x}$, N$_{y}$ and N$_{z}$) respectively on $x, y, z$ axis;
+	\item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments; \RC{CE tu vÃ©rifies, je dis ca de tÃªte}
+	\item matrix off-diagonal value;
+	\item execution mode: synchronous or asynchronous;
+	\RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les  arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramÃ¨tre, on peut ne pas en parler}
+	\item Size of matrix S;
+	\item Maximum number of iterations and tolerance threshold for CGLS.
 \end{itemize}
 
-At last, note that the two solver algorithms have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
@@ -320,103 +365,123 @@ At last, note that the two solver algorithms have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
+In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+
+\subsection{The 3D Poisson problem}
 
-\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.
+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.
 
-\textbf{Step 1} : Choose with the end users the class of algorithms or
+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.
+
+\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. \\
 
-\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 (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 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
+\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 }: Set up the different grid testbed environments
-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 8.10$^{-6}$
-microseconds (resp. 5.10$^{-5}$) for the intra-clusters links (resp.
-inter-clusters backbone links). \\
+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 in varying the key parameters, especially
+within these configurations by varying the key parameters, especially
 the CPU power capacity, the network parameters and also the size of the
-input matrix. Note that some parameters like some program input arguments should be fixed to be invariant to allow the comparison. \\
+input data.  \\
 
 \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 and 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
-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
-\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.
+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  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 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   practice,  these  two   networks  have  different   speeds.
+ The intra-network  generally works  like a  high speed  local network  with a
+ high bandwith and very low latency. In opposite, the inter-network connects
+ clusters sometime via  heterogeneous networks components  throuth internet with
+ a lower speed.  The network  between distant  clusters might  be a  bottleneck
+ for  the global performance of the application.
+
+\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+
+In the scope  of this paper, our  first objective is to analyze  when the Krylov
+Multisplitting  method   has  better  performances  than   the  classical  GMRES
+method. With an  iterative method, better performances mean a  smaller number of
+iterations and execution time before reaching the convergence.  For a systematic
+study,  the experiments  should figure  out  that, for  various grid  parameters
+values, the simulator will confirm  the targeted outcomes, 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
+\subsubsection{Execution of the the algorithms on various computational grid
 architecture and scaling up the input matrix size}
-\\
-
+\ \\
 % environment
-\begin{footnotesize}
+
+\begin{figure} [ht!]
+\begin{center}
 \begin{tabular}{r c }
  \hline
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
@@ -424,47 +489,57 @@ architecture and scaling up the input matrix size}
  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 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la lÃ©gende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}
 
-\end{footnotesize}
 
 
 
 %\RCE{J'ai voulu mettre les tableaux des donnÃ©es mais je pense que c'est inutile et Ã§a va surcharger}
 
 
-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.
+In this  section, we analyze the  performences of algorithms running  on various
+grid configuration  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+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.
+
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vÃ©rifie...}
+\RC{Les lÃ©gendes ne sont pas explicites...}
+
 
-%\begin{wrapfigure}{l}{100mm}
 \begin{figure} [ht!]
-\centering
-\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}}
+  \begin{center}
+    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+  \end{center}
+  \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+  \label{fig:01}
 \end{figure}
-%\end{wrapfigure}
 
-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 two different speed cluster inter-networks\\}
+The execution  times between  the two algorithms  is significant  with different
+grid architectures, even  with the same number of processors  (for example, 2x16
+and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+(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.
 
-% environment
-\begin{footnotesize}
+\subsubsection{Running on two different speed cluster inter-networks}
+\ \\
+
+\begin{figure} [ht!]
+\begin{center}
 \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 \\
+ 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}
+\caption{Clusters x Nodes - Networks N1 x N2}
+\end{center}
+\end{figure}
 
 
 
@@ -473,31 +548,30 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\label{fig:02}
 \end{figure}
 %\end{wrapfigure}
 
-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\\}
+These experiments  compare the  behavior of  the algorithms  running first  on a
+speed inter-cluster  network (N1) and  also on  a less performant  network (N2).
+Figure~\ref{fig:02} shows that end users will  gain to reduce the execution time
+for  both  algorithms  in using  a  grid  architecture  like  4x16 or  8x8:  the
+performance was increased  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\%. \RC{c'est pas clair : la diffÃ©rence entre quoi et quoi?}
 
-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
 \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\\
+ 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}
+\caption{Network latency impact}
+\end{figure}
 
 
 
@@ -505,123 +579,124 @@ Table 3 : Network latency impact \\
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impact on execution time}
-%\label{overflow}}
+\label{fig:03}
 \end{figure}
 
 
-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$^{-6}$.
+According  the results  in  Figure~\ref{fig:03}, a  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. Krylov
+multisplitting)   algorithm.   In   addition,   it  appears   that  the   Krylov
+multisplitting method tolerates  more the network latency variation  with a less
+rate  increase  of  the  execution   time.   Consequently,  in  the  worst  case
+(lat=6.10$^{-5 }$), the  execution time for GMRES is almost  the double than the
+time of the Krylov multisplitting, even  though, the performance was on the same
+order of magnitude with a latency of 8.10$^{-6}$.
 
-\textit{\\3.d Network bandwidth impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
 \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 \\
-
-\end{footnotesize}
+\caption{Network bandwidth impact}
+\end{figure}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
 \caption{Network bandwith impact on execution time}
-%\label{overflow}
+\label{fig:04}
 \end{figure}
 
 
 
-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.
+The results  of increasing  the network  bandwidth show  the improvement  of the
+performance  for   both  algorithms   by  reducing   the  execution   time  (see
+Figure~\ref{fig:04}). However,  in this  case, the Krylov  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}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
 \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 \\
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ 
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Input matrix size impact}
+\end{figure}
 
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impact on execution time}
+\label{fig:05}
 \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\\}
+In these experiments, 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 Figure~\ref{fig:05},  the execution
+time for  both algorithms increases when  the input matrix size  also increases.
+But the interesting results are:
+\begin{enumerate}
+  \item the drastic increase (300 times) \RC{Je ne vois pas cela sur la figure}
+of the  number of  iterations needed  to reach the  convergence for  the classical
+GMRES algorithm when  the matrix size go beyond N$_{x}$=150;
+\item the  classical GMRES execution time  is almost the double  for N$_{x}$=140
+  compared with the Krylov multisplitting method.
+\end{enumerate}
+
+These  findings may  help a  lot end  users to  setup the  best and  the optimal
+targeted environment for the application deployment when focusing on the problem
+size scale up.  It  should be noticed that the same test has  been done with the
+grid 2x16 leading to the same conclusion.
+
+\subsubsection{CPU Power impact on performance}
 
-% environment
-\begin{footnotesize}
+\begin{figure} [ht!]
+\centering
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\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 \\
-
-\end{footnotesize}
-
+\caption{CPU Power impact}
+\end{figure}
 
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
 \caption{CPU Power impact on execution time}
-%\label{overflow}}
+\label{fig:06}
 \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. 
-
-\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. 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}.
+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 Figure~\ref{fig:06}  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 the multisplitting algorithm in asynchronous mode}
+
+The previous paragraphs  put in evidence the interests to  simulate the behavior
+of the application before any deployment in a real environment.  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
@@ -705,8 +780,7 @@ CONCLUSION
 
 \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}
