Desription du problème à résoudre: 3D Poisson equation

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index 8f39683c3c12888c896932a733ab10d210630d9d..95683d5284e523a99084d8e42b321046e4ff197b 100644 (file)
--- 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
 
 
 %% 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
 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.
 
 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
 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


@@ -115,12 +115,12 @@ their  applications  using  a simulation tool before.
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 
 
 %\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
 %performance} 

-\keywords{Multisplitting algorithms, Synchronous and asynchronous iterations, SimGrid, Simulation, Performance evaluation}
+\keywords{ Performance evaluation, Simulation, SimGrid,  Synchronous and asynchronous iterations, Multisplitting algorithms}

 
 \maketitle
 
 
 \maketitle
 

-\section{Introduction}  The use of multi-core architectures for solving large
-scientific problems seems to  become imperative  in  a  lot  of  cases.
+\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.
 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.


@@ -131,8 +131,7 @@ 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,
 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
 
 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


@@ -147,7 +146,7 @@ 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
 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}.
+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
 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


@@ -155,15 +154,15 @@ the asynchronous  iterative scheme  can significantly  reduce  overall execution
 times by  suppressing idle  times due to  synchronizations~(see~\cite{bahi07}
 for more details).
 
 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
+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
 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
 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
 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}
 
 \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}
 
 \section{SimGrid}
  \label{sec:simgrid}
 


@@ -203,12 +223,12 @@ concluding remarks and perspectives.
 \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}
 \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}


@@ -239,12 +259,12 @@ where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are compute
 %\end{algorithm}
 \end{figure}
 
 %\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 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}
 \end{equation}
 \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.
+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
 \begin{equation}
 
 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}


@@ -287,32 +307,48 @@ The algorithm in Figure~\ref{alg:02} includes the procedure of the residual mini
 \subsection{Simulation of two-stage methods using SimGrid framework}
 \label{sec:04.02}
 
 \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
+       \item archi   : grid computational description (number of clusters, number of

 nodes/processors for each cluster).
 \end{itemize}
 nodes/processors for each cluster).
 \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.

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


@@ -320,79 +356,101 @@ At last, note that the two solver algorithms have been executed with the Simgrid
 \section{Experimental Results}
 \label{sec:expe}
 
 \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
+\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. \\
 
 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
+\textbf{Step 2}: Collect the software materials needed for the

 experimentation. In our case, we have two variants algorithms 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. \\
+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
+\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
 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). \\
+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
 
 \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
 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. \\
+input data.  \\

 
 \textbf{Step 6} : Collect and analyze the output results.
 
 \subsection{Factors impacting distributed applications performance in
 a grid environment}
 
 
 \textbf{Step 6} : Collect and analyze the output results.
 
 \subsection{Factors impacting distributed applications performance in
 a grid environment}
 

-From our previous experience on running distributed application in a
-computational grid, many factors are identified to have an impact on the
-program behavior and performance on this specific environment. Mainly,
-first of all, the architecture of the grid itself can obviously
-influence the performance results of the program. The performance gain
-might be important theoretically when the number of clusters and/or the
-number of nodes (processors/cores) in each individual cluster increase.
-
-Another important factor impacting the overall performance of the
-application is the network configuration. Two main network parameters
-can modify drastically the program output results : (i) the network
-bandwidth (bw=bits/s) also known as "the data-carrying capacity"
-of the network is defined as the maximum of data that can pass
-from one point to another in a unit of time. (ii) the network latency
-(lat : microsecond) defined as the delay from the start time to send the
-data from a source and the final time the destination have finished to
-receive it. Upon the network characteristics, another impacting factor
-is the application dependent volume of data exchanged between the nodes
-in the cluster and between distant clusters. Large volume of data can be
-transferred in transit between the clusters and nodes during the code
-execution.
-
- In a grid environment, it is common to distinguish in one hand, the
-"\,intra-network" which refers to the links between nodes within a
-cluster and in the other hand, the "\,inter-network" which is the
-backbone link between clusters. By design, these two networks perform
-with different speed. The intra-network generally works like a high
-speed local network with a high bandwith and very low latency. In
-opposite, the inter-network connects clusters sometime via heterogeneous
-networks components thru internet with a lower speed. The network
-between distant clusters might be a bottleneck for the global
-performance of the application.
+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}
 
 \subsection{Comparing GMRES and Multisplitting algorithms in
 synchronous mode}


@@ -400,7 +458,7 @@ 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
 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
+\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
 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


@@ -413,7 +471,7 @@ and our comments.\\
 
 
 \textit{3.a Executing the algorithms on various computational grid
 
 
 \textit{3.a Executing the algorithms on various computational grid

-architecture scaling up the input matrix size}
+architecture and scaling up the input matrix size}

 \\
 
 % environment
 \\
 
 % environment


@@ -434,8 +492,7 @@ Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
 %\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 3 show the non-variation of the number of
-iterations of classical GMRES for a given input matrix size; it is not
+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}
 the case for the multisplitting method.
 
 %\begin{wrapfigure}{l}{100mm}


@@ -447,15 +504,13 @@ the case for the multisplitting method.
 \end{figure}
 %\end{wrapfigure}
 
 \end{figure}
 %\end{wrapfigure}
 

-Unless the 8x8 cluster, the time
-execution difference between the two algorithms is important when
+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
 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.
+(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}
 
 % environment
 \begin{footnotesize}


@@ -482,11 +537,11 @@ Table 2 : Clusters x Nodes - Networks N1 x N2 \\
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on

-a speed inter- cluster network (N1) and a less performant network (N2).
+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
 Figure 4 shows that end users will gain to reduce the execution time
 for both algorithms in using a grid architecture like 4x16 or 8x8: the
 performance was increased in a factor of 2. The results depict also that

-when the network speed drops down, the difference between the execution
+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\\}
 times can reach more than 25\%.
 
 \textit{\\3.c Network latency impacts on performance\\}


@@ -547,11 +602,8 @@ 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 (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 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\\}
 
 
 \textit{\\3.e Input matrix size impacts on performance\\}
 


@@ -579,7 +631,7 @@ 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
 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
+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
 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


@@ -596,7 +648,7 @@ same test has been done with the grid 2x16 getting the same conclusion.
 \begin{tabular}{r c }
  \hline
  Grid & 2x16\\ %\hline
 \begin{tabular}{r c }
  \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 \\


@@ -609,14 +661,13 @@ 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
+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,
 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.
+after adding more powerful CPU. 

 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}
 
 \subsection{Comparing GMRES in native synchronous mode and
 Multisplitting algorithms in asynchronous mode}


@@ -624,12 +675,12 @@ 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
 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.
+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
 
 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


@@ -639,11 +690,10 @@ 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.
 
 asynchronous may theoretically reduce the overall execution time and can
 improve the algorithm performance.
 

-As stated supra, SIMGRID simulator tool has been used to prove the
+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,
 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.
+\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 : \\


@@ -653,11 +703,11 @@ The test conditions are summarized in the table below : \\
 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline
 \begin{tabular}{r c }
  \hline
  Grid & 2x50 totaling 100 processors\\ %\hline

- Processors & 1 GFlops to 1.5 GFlops\\
-   Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
-   Inter-Network & bw=5 Mbits - lat=2E-02\\
+ 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}
 


@@ -665,12 +715,9 @@ 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
 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.
+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}}


@@ -680,55 +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}


@@ -739,8 +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}