Premier draft de l'intro

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -52,7 +52,12 @@
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \algnewcommand\algorithmicoutput{\textbf{Output:}}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 

-\newcommand{\MI}{\mathit{MaxIter}}
+\newcommand{\TOLG}{\mathit{tol_{gmres}}}
+\newcommand{\MIG}{\mathit{maxit_{gmres}}}
+\newcommand{\TOLM}{\mathit{tol_{multi}}}
+\newcommand{\MIM}{\mathit{maxit_{multi}}}
+\newcommand{\TOLC}{\mathit{tol_{cgls}}}
+\newcommand{\MIC}{\mathit{maxit_{cgls}}}

 
 \usepackage{array}
 \usepackage{color, colortbl}
 
 \usepackage{array}
 \usepackage{color, colortbl}


@@ -87,14 +92,68 @@
 %% 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}
 %% 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}

-ABSTRACT
+  The behavior of multicore applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications. We have decided to use SimGrid as it enables to benchmark MPI applications.
+
+In this paper, we focus our attention on two parallel iterative algorithms based
+on the  Multisplitting algorithm  and we  compare them  to the  GMRES algorithm.
+These algorithms  are used to  solve libear  systems. Two different  variantsof the Multisplitting are
+studied: one  using synchronoous  iterations and  another one  with asynchronous
+iterations. For each algorithm we have  tested different parameters to see their
+influence.  We strongly  recommend people  interested  by investing  into a  new
+expensive  hardware  architecture  to   benchmark  their  applications  using  a
+simulation tool before.
+
+  
+
+  

 \end{abstract}
 
 \keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
 
 \maketitle
 
 \end{abstract}
 
 \keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
 
 \maketitle
 

-\section{Introduction}
+\section{Introduction} 
+The use of multi-core architectures for solving large scientific problems seems to  become imperative  in  a  lot  of  cases. 
+Whatever the scale of these architectures (distributed clusters, computational grids, embedded multi-core \ldots) they  are generally 
+well adapted to execute complexe parallel applications operating on a large amount of data.  Unfortunately,  users (industrials or scientists), 
+who need such computational resources may not have an easy access to such efficient architectures. The cost of using the platform and/or the cost of 
+testing and deploying an application are often very important.  So, in this context it is difficult to optimize a given application for a given 
+architecture. In this way and in order to reduce the access cost to these computing resources it seems very interesting to use a simulation environment. 
+The advantages are numerous: development life cycle, code debugging, ability to obtain results quickly \ldots
+
+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.  Then each processing unit has to
+compute an iteration, to send/receive some data dependencies to/from
+its neighbors and to iterate this process until the convergence of
+the method. Several well-known methods demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.
+In this processing mode a task cannot begin a new iteration while it
+has not received data dependencies from its neighbors. We say that the iteration computation follows a synchronous scheme.
+In the asynchronous scheme a task can compute a new iteration without having to
+wait for the data dependencies coming from its neighbors. Both
+communication and computations are asynchronous inducing that there is
+no more idle times, due to synchronizations, between two
+iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks that we detail in section 2 but even if the number of iterations required to converge is 
+generally  greater  than for the synchronous  case, it appears that the asynchronous  iterative scheme  can significantly  reduce  overall execution
+times by  suppressing idle  times due to  synchronizations~\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 leverage the possibility of testing various platform scenarios.
+
+The main contribution of this paper is to show that the use of a simulation tool (i.e. the SimGrid   toolkit~\cite{SimGrid}) in the context of real 
+parallel applications (i.e. large linear system solver) can help developers to better tune their application for a given multi-core architecture.
+To show the validity of this approach we first compare the simulated execution of the multisplitting algorithm  with  the  GMRES   (Generalized   Minimal  Residual) solver
+\cite{ref1} both in synchronous mode. The obtained results on different simulated multi-core architectures confirm the results previously obtained on non simulated architecture. 
+We also confirm  the efficiency  of the  asynchronous  multisplitting algorithm  comparing to the synchronous  GMRES. In this way and with a simple computing architecture (a laptop) 
+SimGrid allows us (with small modifications of the MPI code) to run a test campaign  of  a  real parallel iterative  applications on  different simulated multi-core architectures. 
+To our knowledge, there is no related work on the large-scale multi-core simulation of a real synchronous and asynchronous iterative application.  
+
+This paper is organized as follows:
+

 
 \section{The asynchronous iteration model}
 
 
 \section{The asynchronous iteration model}
 


@@ -103,72 +162,146 @@ ABSTRACT
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%
 

-\section{Two-stage splitting methods}
+\section{Two-stage multisplitting methods}

 \label{sec:04}
 \label{sec:04}

-\subsection{Multisplitting methods for sparse linear systems}
+\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}

 \label{sec:04.01}
 \label{sec:04.01}

-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. The 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). The two-stage multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows

 \begin{equation}
 \begin{equation}

-x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell M^{-1}_\ell (N_\ell x^k + b),~k=1,2,3,\ldots
+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}
 \label{eq:02}
 \end{equation}

-where a collection of $L$ triplets $(M_\ell, N_\ell, E_\ell)$ defines the multisplitting of matrix $A$, such that: the different splittings are defined as $A=M_\ell-N_\ell$ where $M_\ell$ are nonsingular matrices, and $\sum_\ell{E_\ell=I}$ are diagonal nonnegative weighting matrices and $I$ is the identity matrix. The iterations of the multisplitting methods can naturally be computed in parallel such that each processor or a group 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}
 \begin{equation}

-M_\ell y_\ell^{k+1} = R_\ell^k,\mbox{~such that~} R_\ell^k = N_\ell x^k_\ell + b,
+A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,

 \label{eq:03}
 \end{equation}
 \label{eq:03}
 \end{equation}

-then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01})
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}. 
+
+\begin{figure}[t]
+%\begin{algorithm}[t]
+%\caption{Block Jacobi two-stage multisplitting method}
+\begin{algorithmic}[1]
+  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+    \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\label{solve}
+    \State Send $x_\ell^k$ to neighboring clusters\label{send}
+    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters\label{recv}
+  \EndFor
+\end{algorithmic}
+\caption{Block Jacobi two-stage multisplitting method}
+\label{alg:01}
+%\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

 \begin{equation}
 \begin{equation}

-x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y^{k+1}_\ell.
+k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,

 \label{eq:04}
 \label{eq:04}

-\end{equation}
-The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors. It is dependent on the condition  
+\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 

 \begin{equation}
 \begin{equation}

-\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1,
+S=[x^1,x^2,\ldots,x^s],~s\ll n.

 \label{eq:05}
 \label{eq:05}

-\end{equation}
-where $\rho$ is the spectral radius of the square matrix. The different linear splittings~(\ref{eq:03}) arising from the multisplitting of matrix $A$can be solved exactly with a direct method or approximated with an iterative method. When the inner method used to solve the linear sub-systems is iterative, the multisplitting method is called {\it inner-outer iterative method} or {\it two-stage multisplitting method}.
-
-In this paper we are focused on two-stage multisplitting methods where the well-known iterative method GMRES is used as an inner iteration. 
+\end{equation}   
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+\begin{equation}
+\min_{\alpha\in\mathbb{R}^s}{\|b-AS\alpha\|_2}.
+\label{eq:06}
+\end{equation} 
+The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
+ 
+\begin{figure}[t]
+%\begin{algorithm}[t]
+%\caption{Krylov two-stage method using block Jacobi multisplitting}
+\begin{algorithmic}[1]
+  \Input $A_\ell$ (sparse matrix), $b_\ell$ (right-hand side)
+  \Output $x_\ell$ (solution vector)\vspace{0.2cm}
+  \State Set the initial guess $x^0$
+  \For {$k=1,2,3,\ldots$ until convergence}
+    \State $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m^{k-1}$
+    \State $x^k_\ell=Solve_{gmres}(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$
+    \State $S_{\ell,k\mod s}=x_\ell^k$
+    \If{$k\mod s = 0$}
+       \State $\alpha = Solve_{cgls}(AS,b,\MIC,\TOLC)$\label{cgls}
+       \State $\tilde{x_\ell}=S_\ell\alpha$
+       \State Send $\tilde{x_\ell}$ to neighboring clusters
+       \Else 
+         \State Send $x_\ell^k$ to neighboring clusters
+    \EndIf
+    \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters
+  \EndFor
+\end{algorithmic}
+\caption{Krylov two-stage method using block Jacobi multisplitting}
+\label{alg:02}
+%\end{algorithm} 
+\end{figure}

 
 \subsection{Simulation of two-stage methods using SimGrid framework}
 
 \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}
+
+\begin{itemize}
+       \item hostfile: Hosts description file.
+       \item plarform: File describing the platform architecture : clusters (CPU power,
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+latency lat, \dots{}).
+       \item archi   : Grid computational description (Number of clusters, Number of 
+nodes/processors for each cluster). 
+\end{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.
+\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.  

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

-\section{Experimental, Results and Comments}
+\section{Experimental Results}

 
 
 
 

-\textbf{V.1. Setup study and Methodology}
+\subsection{Setup study and Methodology}

 
 To conduct our study, we have put in place the following methodology 
 
 To conduct our study, we have put in place the following methodology 

-which can be reused with any grid-enabled applications.
+which can be reused for any grid-enabled applications.

 
 \textbf{Step 1} : Choose with the end users the class of algorithms or 
 the application to be tested. Numerical parallel iterative algorithms 
 
 \textbf{Step 1} : Choose with the end users the class of algorithms or 
 the application to be tested. Numerical parallel iterative algorithms 

-have been chosen for the study in the paper. 
+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 three variants algorithms for the 
-resolution of three 3D-Poisson problem: (1) using the classical GMRES 
-\textit{(Generalized Minimal RESidual Method)} alias Algo-1 in this 
-paper, (2) using the multisplitting method alias Algo-2 and (3) an 
-enhanced version of the multisplitting method as Algo-3. 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 $[$10$]$ but also in a virtual 
-machine on a laptop.
+experimentation. In our case, we have two variants algorithms for the 
+resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, SIMGRID simulator has been chosen to simulate the behaviors of the 
+distributed applications. SIMGRID is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\

 
 \textbf{Step 3} : Fix the criteria which will be used for the future 
 results comparison and analysis. In the scope of this study, we retain 
 in one hand the algorithm execution mode (synchronous and asynchronous) 
 and in the other hand the execution time and the number of iterations of 
 
 \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.
+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 
 
 \textbf{Step 4 }: Setup up the different grid testbeds environment 
 which will be simulated in the simulator tool to run the program. The 


@@ -177,17 +310,17 @@ 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. 
 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).
+inter-clusters backbone links). \\

 
 

-\textbf{Step 5}: Process an extensive and comprehensive testings 
+\textbf{Step 5}: Conduct an extensive and comprehensive testings 

 within these configurations in varying the key parameters, especially 
 the CPU power capacity, the network parameters and also the size of the 
 within these configurations in varying the key parameters, especially 
 the CPU power capacity, the network parameters and also the size of the 

-input matrix. Note that some parameters should be invariant to allow the 
-comparison like some program input arguments.
+input matrix. Note that some parameters should be fixed to be invariant to allow the 
+comparison like some program input arguments. \\

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

-\textbf{ V.2. Factors impacting distributed applications performance in 
+\subsection{Factors impacting distributed applications performance in 

 a grid environment}
 
 From our previous experience on running distributed application in a 
 a grid environment}
 
 From our previous experience on running distributed application in a 


@@ -202,7 +335,7 @@ 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" 
 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" 

-$[$13$]$ of the network is defined as the maximum of data that can pass 
+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 
 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 


@@ -223,22 +356,22 @@ networks components thru internet with a lower speed. The network
 between distant clusters might be a bottleneck for the global 
 performance of the application. 
 
 between distant clusters might be a bottleneck for the global 
 performance of the application. 
 

-\textbf{V.3 Comparing GMRES and Multisplitting algorithms in 
+\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 
 \textbf{\textit{synchronous mode}}. Better algorithm performance 
 synchronous mode}
 
 In the scope of this paper, our first objective is to demonstrate the 
 Algo-2 (Multisplitting method) shows a better performance in grid 
 architecture compared with Algo-1 (Classical GMRES) both running in 
 \textbf{\textit{synchronous mode}}. Better algorithm performance 

-should mean a less number of iterations output and a less execution time 
+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 
 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 $[$12$]$.
+on the chosen class of algorithm.

 
 The following paragraphs present the test conditions, the output results 
 
 The following paragraphs present the test conditions, the output results 

-and our comments.
+and our comments.\\

 
 
 \textit{3.a Executing the algorithms on various computational grid 
 
 
 \textit{3.a Executing the algorithms on various computational grid 


@@ -250,27 +383,28 @@ architecture scaling up the input matrix size}
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline

- Network & N2 : bw=1Gbs-lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
- - & N$_{x}$ =170 x 170 x 170    \\ \hline
+ Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
+ - &  N$_{x}$ x N$_{y}$ x N$_{z}$  =170 x 170 x 170    \\ \hline

  \end{tabular}
  \end{tabular}

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

 
 

+\end{footnotesize}

 
 

- Table 1 : Clusters x Nodes with NX=150 or NX=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 1 show the non-variation of the number of 
+The results in figure 3 show the non-variation of the number of 

 iterations of classical GMRES for a given input matrix size; it is not 
 the case for the multisplitting method. 
 
 iterations of classical GMRES for a given input matrix size; it is not 
 the case for the multisplitting method. 
 

-%\begin{wrapfigure}{l}{60mm}
+%\begin{wrapfigure}{l}{100mm}

 \begin{figure} [ht!]
 \centering
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes NX=150 and NX=170} 
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170} 

 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}
 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}


@@ -283,41 +417,41 @@ 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 to higher input 
 matrix size. 
 

-\textit{3.b Running on various computational grid architecture}
+\textit{\\3.b Running on various computational grid architecture\\}

 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16, 4x8\\ %\hline
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16, 4x8\\ %\hline

- Network & N1 : bw=10Gbs-lat=8E-06 \\ %\hline
- - & N2 : bw=1Gbs-lat=5E-05 \\
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
+ Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+ - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\

  \end{tabular}
  \end{tabular}

-\end{footnotesize}
+Table 2 : Clusters x Nodes - Networks N1 x N2 \\
+
+ \end{footnotesize}

 
 

-%Table 2 : Clusters x Nodes - Networks N1 x N2
-%\RCE{idem pour tous les tableaux de donnees}

 
 
 
 

-%\begin{wrapfigure}{l}{60mm}
+%\begin{wrapfigure}{l}{100mm}

 \begin{figure} [ht!]
 \centering
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{cluster_x_nodes_n1_x_n2.pdf}
+\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}

 \caption{Cluster x Nodes N1 x N2}
 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on 
 \caption{Cluster x Nodes N1 x N2}
 %\label{overflow}}
 \end{figure}
 %\end{wrapfigure}
 
 The experiments compare the behavior of the algorithms running first on 

-speed inter- cluster network (N1) and a less performant network (N2). 
-The figure 2 shows that end users will gain to reduce the execution time 
+a speed inter- cluster network (N1) and a less performant network (N2). 
+Figure 4 shows that end users will gain to reduce the execution time 

 for both algorithms in using a grid architecture like 4x16 or 8x8: the 
 performance was increased in a factor of 2. The results depict also that 
 when the network speed drops down, the difference between the execution 
 times can reach more than 25\%. 
 
 for both algorithms in using a grid architecture like 4x16 or 8x8: the 
 performance was increased in a factor of 2. The results depict also that 
 when the network speed drops down, the difference between the execution 
 times can reach more than 25\%. 
 

-\textit{\\\\\\\\\\\\\\\\\\3.c Network latency impacts on performance}
+\textit{\\3.c Network latency impacts on performance\\}

 
 % environment
 \begin{footnotesize}
 
 % environment
 \begin{footnotesize}


@@ -325,48 +459,50 @@ times can reach more than 25\%.
  \hline  
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  \hline  
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline

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

  \end{tabular}
  \end{tabular}

+Table 3 : Network latency impact \\
+

 \end{footnotesize}
 
 \end{footnotesize}
 

-Table 3 : Network latency impact

 
 
 \begin{figure} [ht!]
 \centering
 
 
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}

 \caption{Network latency impact on execution time}
 %\label{overflow}}
 \end{figure}
 
 
 \caption{Network latency impact on execution time}
 %\label{overflow}}
 \end{figure}
 
 

-According the results in table and figure 3, degradation of the network 
+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 
 latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time 
 increase more than 75\% (resp. 82\%) of the execution for the classical 
 GMRES (resp. multisplitting) algorithm. In addition, it appears that the 
 multisplitting method tolerates more the network latency variation with 

-a less rate increase. Consequently, in the worst case (lat=6.10$^{-5
+a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-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}$. 
 
 }$), the execution time for GMRES is almost the double of the time for 
 the multisplitting, even though, the performance was on the same order 
 of magnitude with a latency of 8.10$^{-6}$. 
 

-\textit{3.d Network bandwidth impacts on performance}
+\textit{\\3.d Network bandwidth impacts on performance\\}

 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16\\ %\hline
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 2x16\\ %\hline

- Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
+ Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
+ Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\

  \end{tabular}
  \end{tabular}

+Table 4 : Network bandwidth impact \\
+

 \end{footnotesize}
 
 \end{footnotesize}
 

-Table 4 : Network bandwidth impact

 
 \begin{figure} [ht!]
 \centering
 
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}

 \caption{Network bandwith impact on execution time}
 %\label{overflow}
 \end{figure}
 \caption{Network bandwith impact on execution time}
 %\label{overflow}
 \end{figure}


@@ -375,46 +511,47 @@ 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 
 
 The results of increasing the network bandwidth depict the improvement 
 of the performance by reducing the execution time for both of the two 

-algorithms. However, and again in this case, the multisplitting method 
+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.
 
 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\\}

 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 4x8\\ %\hline
 
 % environment
 \begin{footnotesize}
 \begin{tabular}{r c }
  \hline  
  Grid & 4x8\\ %\hline

- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\

  \end{tabular}
  \end{tabular}

+Table 5 : Input matrix size impact\\
+

 \end{footnotesize}
 
 \end{footnotesize}
 

-Table 5 : Input matrix size impact

 
 \begin{figure} [ht!]
 \centering
 
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{pb_size_impact_on_execution_time.pdf}
+\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}

 \caption{Pb size impact on execution time}
 %\label{overflow}}
 \end{figure}
 
 In this experimentation, the input matrix size has been set from 
 \caption{Pb size impact on execution time}
 %\label{overflow}}
 \end{figure}
 
 In this experimentation, the input matrix size has been set from 

-Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
-200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 5, 
-the execution time for the algorithms convergence increases with the 
-input matrix size. But the interesting result here direct on (i) the 
+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 
 drastic increase (300 times) of the number of iterations needed before 
 the convergence for the classical GMRES algorithm when the matrix size 

-go beyond Nx=150; (ii) the classical GMRES execution time also almost 
-the double from Nx=140 compared with the convergence time of the 
+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.
 
 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}
+\textit{\\3.f CPU Power impact on performance\\}

 
 % environment
 \begin{footnotesize}
 
 % environment
 \begin{footnotesize}


@@ -424,13 +561,14 @@ same test has been done with the grid 2x16 getting the same conclusion.
  Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}
  Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
  \end{tabular}

+Table 6 : CPU Power impact \\
+

 \end{footnotesize}
 
 \end{footnotesize}
 

-Table 6 : CPU Power impact

 
 \begin{figure} [ht!]
 \centering
 
 \begin{figure} [ht!]
 \centering

-\includegraphics[width=60mm]{cpu_power_impact_on_execution_time.pdf}
+\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}

 \caption{CPU Power impact on execution time}
 %\label{overflow}}
 \end{figure}
 \caption{CPU Power impact on execution time}
 %\label{overflow}}
 \end{figure}


@@ -442,7 +580,7 @@ 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. Note that the execution time axis in the 
 figure is in logarithmic scale.
 

- \textbf{V.4 Comparing GMRES in native synchronous mode and 
+\subsection{Comparing GMRES in native synchronous mode and 

 Multisplitting algorithms in asynchronous mode}
 
 The previous paragraphs put in evidence the interests to simulate the 
 Multisplitting algorithms in asynchronous mode}
 
 The previous paragraphs put in evidence the interests to simulate the 


@@ -470,7 +608,7 @@ best combination of the grid resources (CPU, Network, input matrix size,
 classical GMRES time. 
 
 
 classical GMRES time. 
 
 

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

 
 % environment
 \begin{footnotesize}
 
 % environment
 \begin{footnotesize}


@@ -481,7 +619,7 @@ The test conditions are summarized in the table below :
    Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2E-02\\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
    Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2E-02\\
  Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline

- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline
+ Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline \\

  \end{tabular}
 \end{footnotesize}
 
  \end{tabular}
 \end{footnotesize}
 


@@ -508,27 +646,27 @@ internet.
   \centering
   \caption{Relative gain of the multisplitting algorithm compared with 
 the classical GMRES}
   \centering
   \caption{Relative gain of the multisplitting algorithm compared with 
 the classical GMRES}

-  \label{tab.cluster.2x50}
+  \label{"Table 7"}

 
   \begin{mytable}{6}
     \hline
 
   \begin{mytable}{6}
     \hline

-    bw
-    & 5         & 5         & 5         & 5         & 5         & 50 \\
+    bandwidth (Mbit/s)
+    & 5         & 5         & 5         & 5         & 5 \\

     \hline
     \hline

-    lat
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
+    latency (ms)
+    & 20      & 20      & 20      & 20      & 20 \\

     \hline
     \hline

-    power
-    & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
+    power (GFlops)
+    & 1         & 1         & 1         & 1.5       & 1.5 \\

     \hline
     \hline

-    size
-    & 62        & 62        & 62        & 100       & 100       & 110 \\
+    size (N)
+    & 62        & 62        & 62        & 100       & 100 \\

     \hline
     \hline

-    Prec/Eprec
-    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} & \np{E-11} \\
+    Precision
+    & \np{E-5}  & \np{E-8}  & \np{E-9}  & \np{E-11} & \np{E-11} \\

     \hline
     \hline

-    speedup
-    & 0.396     & 0.392     & 0.396     & 0.391     & 0.393     & 0.395 \\
+    Relative gain
+    & 2.52     & 2.55     & 2.52     & 2.57     & 2.54 \\

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


@@ -536,23 +674,23 @@ the classical GMRES}
 
   \begin{mytable}{6}
     \hline
 
   \begin{mytable}{6}
     \hline

-    bw
-    & 50        & 50        & 50        & 50        & 10        & 10 \\
+    bandwidth (Mbit/s)
+    & 50        & 50        & 50        & 50        & 50 \\

     \hline
     \hline

-    lat
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
+    latency (ms)
+    & 20      & 20      & 20      & 20      & 20 \\

     \hline
     \hline

-    power
-    & 1.5       & 1.5       & 1.5       & 1.5       & 1         & 1.5 \\
+    power (GFlops)
+    & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\

     \hline
     \hline

-    size
-    & 120       & 130       & 140       & 150       & 171       & 171 \\
+    size (N)
+    & 110       & 120       & 130       & 140       & 150 \\

     \hline
     \hline

-    Prec/Eprec
-    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5}  & \np{E-5} \\
+    Precision
+    & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\

     \hline
     \hline

-    speedup
-    & 0.398     & 0.388     & 0.393     & 0.394     & 0.63      & 0.778 \\
+    Relative gain
+    & 2.53     & 2.51     & 2.58     & 2.55     & 2.54 \\

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


@@ -567,12 +705,8 @@ CONCLUSION
 The authors would like to thank\dots{}
 
 
 The authors would like to thank\dots{}
 
 

-% trigger a \newpage just before the given reference
-% number - used to balance the columns on the last page
-% adjust value as needed - may need to be readjusted if
-% the document is modified later
-\bibliographystyle{IEEEtran}
-\bibliography{hpccBib}
+\bibliographystyle{wileyj}
+\bibliography{biblio}

 
 \end{document}
 
 
 \end{document}