X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/f463119047e7f3b45c777e90a8702e74aba0f520..c6ed6c1a3600ab1fed77636cbae31b12198c6485:/paper.tex?ds=inline

diff --git a/paper.tex b/paper.tex
index f3ef835..eaaa8fb 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,13 @@
-\documentclass[conference]{IEEEtran}
+\documentclass[times]{cpeauth}
+
+\usepackage{moreverb}
+
+%\usepackage[dvips,colorlinks,bookmarksopen,bookmarksnumbered,citecolor=red,urlcolor=red]{hyperref}
+
+%\newcommand\BibTeX{{\rmfamily B\kern-.05em \textsc{i\kern-.025em b}\kern-.08em
+%T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+
+\def\volumeyear{2015}
 
 \usepackage{graphicx}
 \usepackage{wrapfig}
@@ -27,6 +36,7 @@
 
 \usepackage{xspace}
 \usepackage[textsize=footnotesize]{todonotes}
+
 \newcommand{\AG}[2][inline]{%
   \todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
 \newcommand{\RC}[2][inline]{%
@@ -42,7 +52,12 @@
 \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}
@@ -53,49 +68,181 @@
 \definecolor{Gray}{gray}{0.9}
 
 
+
 \begin{document}
 \RCE{Titre a confirmer.}
-
 \title{Comparative performance analysis of simulated grid-enabled numerical iterative algorithms}
+%\itshape{\journalnamelc}\footnotemark[2]}
 
-\author{%
-  \IEEEauthorblockN{%
-    Charles Emile Ramamonjisoa and
+\author{    Charles Emile Ramamonjisoa and
     David Laiymani and
     Arnaud Giersch and
     Lilia Ziane Khodja and
     RaphaÃ«l Couturier
-  }
-  \IEEEauthorblockA{%
+}
+
+\address{
+	\centering    
     Femto-ST Institute - DISC Department\\
     UniversitÃ© de Franche-ComtÃ©\\
     Belfort\\
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
-  }
 }
 
-\maketitle
+%% 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.
 
-Keywords : Algorithm distributed iterative asynchronous simulation simgrid performance
+  
 
+  
 \end{abstract}
 
-\section{Introduction}
+\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
+
+\maketitle
+
+\section{Introduction} 
 
 \section{The asynchronous iteration model}
 
 \section{SimGrid}
 
-\section{Simulation of the multisplitting method}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage multisplitting methods}
+\label{sec:04}
+\subsection{Synchronous and asynchronous two-stage methods for sparse linear systems}
+\label{sec:04.01}
+In this paper we focus on two-stage multisplitting methods in their both versions synchronous and asynchronous~\cite{Frommer92,Szyld92,Bru95}. These iterative methods are based on multisplitting methods~\cite{O'leary85,White86,Alefeld97} and use two nested iterations: the outer iteration and the inner iteration. Let us consider the following sparse linear system of $n$ equations in $\mathbb{R}$
+\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
+\begin{equation}
+x_\ell^{k+1} = A_{\ell\ell}^{-1}(b_\ell - \displaystyle\sum^{L}_{\substack{m=1\\m\neq\ell}}{A_{\ell m}x^k_m}),\mbox{~for~}\ell=1,\ldots,L\mbox{~and~}k=1,2,3,\ldots
+\label{eq:02}
+\end{equation}
+where $x_\ell$ are sub-vectors of the solution $x$, $b_\ell$ are the sub-vectors of the right-hand side $b$, and $A_{\ell\ell}$ and $A_{\ell m}$ are diagonal and off-diagonal blocks of matrix $A$ respectively. The iterations of these methods can naturally be computed in parallel such that each processor or cluster of processors is responsible for solving one splitting as a linear sub-system 
+\begin{equation}
+A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
+\label{eq:03}
+\end{equation}
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Bru95,bahi07}. 
+
+\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}
+k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
+\label{eq:04}
+\end{equation}    
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm. 
+
+The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration 
+\begin{equation}
+S=[x^1,x^2,\ldots,x^s],~s\ll n.
+\label{eq:05}
+\end{equation}   
+At each $s$ outer iterations, the algorithm computes a new approximation $\tilde{x}=S\alpha$ which minimizes the residual
+\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}
+\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 PLATFORM: File describing the platform architecture : clusters (CPU power,
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+lat latency, \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 (NX, NY and NZ);
+	\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}
 
 
-\textbf{V.1. Setup study and Methodology}
+\subsection{Setup study and Methodology}
 
 To conduct our study, we have put in place the following methodology 
 which can be reused with any grid-enabled applications.
@@ -106,13 +253,12 @@ have been chosen for the study in the paper.
 
 \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 
+resolution of three 3D-Poisson problem: (1) using the classical GMRES 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 
+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 
@@ -136,9 +282,9 @@ 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.
 
-\textbf{Step 6} : Collect and analyze the output results.
+{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 
@@ -153,7 +299,7 @@ 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" 
-$[$13$]$ 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 
@@ -174,7 +320,7 @@ networks components thru internet with a lower speed. The network
 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 
@@ -186,7 +332,7 @@ 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 
 and our comments.
@@ -194,7 +340,7 @@ and our comments.
 
 \textit{3.a Executing the algorithms on various computational grid 
 architecture scaling up the input matrix size}
-
+\\
 
 % environment
 \begin{footnotesize}
@@ -205,22 +351,29 @@ architecture scaling up the input matrix size}
  Input matrix size & N$_{x}$ =150 x 150 x 150 and\\ %\hline
  - & N$_{x}$ =170 x 170 x 170    \\ \hline
  \end{tabular}
-\end{footnotesize}
+Table 1 : Clusters x Nodes with NX=150 or NX=170 \\
 
+\end{footnotesize}
 
- Table 1 : Clusters x Nodes with NX=150 or NX=170
-\RCE{J'ai voulu mettre les tableaux des donnÃ©es mais je pense que c'est inutile et Ã§a va surcharger}
 
-\begin{wrapfigure}{l}{50mm}
-\centering
-\includegraphics[width=50mm]{Cluster x Nodes NX=150 and NX=170.jpg}
-\caption{Cluster x Nodes NX=150 and NX=170 \label{overflow}}
-\end{wrapfigure}
+ 
+%\RCE{J'ai voulu mettre les tableaux des donnÃ©es mais je pense que c'est inutile et Ã§a va surcharger}
 
 
 The results in figure 1 show the non-variation of the number of 
 iterations of classical GMRES for a given input matrix size; it is not 
-the case for the multisplitting method. Unless the 8x8 cluster, the time 
+the case for the multisplitting method. 
+
+%\begin{wrapfigure}{l}{100mm}
+\begin{figure} [ht!]
+\centering
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\caption{Cluster x Nodes NX=150 and NX=170} 
+%\label{overflow}}
+\end{figure}
+%\end{wrapfigure}
+
+Unless the 8x8 cluster, the time 
 execution difference between the two algorithms is important when 
 comparing between different grid architectures, even with the same number of 
 processors (like 2x16 and 4x8 = 32 processors for example). The 
@@ -228,7 +381,7 @@ experiment concludes the low sensitivity of the multisplitting method
 (compared with the classical GMRES) when scaling up to higher input 
 matrix size. 
 
-\textit{3.b Running on various computational grid architecture}
+\textit{\\3.b Running on various computational grid architecture\\}
 
 % environment
 \begin{footnotesize}
@@ -237,19 +390,22 @@ matrix size.
  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
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline \\
  \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}{45mm}
+%\begin{wrapfigure}{l}{100mm}
+\begin{figure} [ht!]
 \centering
-\includegraphics[width=50mm]{Cluster x Nodes N1 x N2.jpg}
-\caption{Cluster x Nodes N1 x N2\label{overflow}}
-\end{wrapfigure}
+\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 
 speed inter- cluster network (N1) and a less performant network (N2). 
@@ -259,7 +415,7 @@ 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}
@@ -267,18 +423,21 @@ times can reach more than 25\%.
  \hline  
  Grid & 2x16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
+ Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline\\
  \end{tabular}
+
+Table 3 : Network latency impact \\
+
 \end{footnotesize}
 
-Table 3 : Network latency impact
 
 
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Network latency impact on execution time.jpg}
-\caption{Network latency impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impact on execution time}
+%\label{overflow}}
+\end{figure}
 
 
 According the results in table and figure 3, degradation of the network 
@@ -291,7 +450,7 @@ a less rate increase. Consequently, in the worst case (lat=6.10$^{-5
 the multisplitting, even though, the performance was on the same order 
 of magnitude with a latency of 8.10$^{-6}$. 
 
-\textit{3.d Network bandwidth impacts on performance}
+\textit{\\3.d Network bandwidth impacts on performance\\}
 
 % environment
 \begin{footnotesize}
@@ -301,15 +460,18 @@ of magnitude with a latency of 8.10$^{-6}$.
  Network & N1 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ =150 x 150 x 150\\ \hline
  \end{tabular}
+
+Table 4 : Network bandwidth impact \\
+
 \end{footnotesize}
 
-Table 4 : Network bandwidth impact
 
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Network bandwith impact on execution time.jpg}
-\caption{Network bandwith impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impact on execution time}
+%\label{overflow}
+\end{figure}
 
 
 
@@ -319,7 +481,7 @@ algorithms. However, and again in this case, the multisplitting method
 presents a better performance in the considered bandwidth interval with 
 a gain of 40\% which is only around 24\% for classical GMRES.
 
-\textit{3.e Input matrix size impacts on performance}
+\textit{\\3.e Input matrix size impacts on performance\\}
 
 % environment
 \begin{footnotesize}
@@ -329,15 +491,17 @@ a gain of 40\% which is only around 24\% for classical GMRES.
  Network & N2 : bw=1Gbs - lat=5E-05 \\ %\hline
  Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
+Table 5 : Input matrix size impact\\
+
 \end{footnotesize}
 
-Table 5 : Input matrix size impact
 
-\begin{wrapfigure}{l}{50mm}
+\begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Pb size impact on execution time.jpg}
-\caption{Pb size impact on execution time\label{overflow}}
-\end{wrapfigure}
+\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 
 Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
@@ -353,7 +517,7 @@ 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}
@@ -363,15 +527,17 @@ 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}
+Table 6 : CPU Power impact \\
+
 \end{footnotesize}
 
-Table 6 : CPU Power impact
 
-\begin{wrapfigure}{l}{60mm}
+\begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{CPU Power impact on execution time.jpg}
-\caption{CPU Power impact on execution time\label{overflow}}
-\end{wrapfigure}
+\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
+\caption{CPU Power impact on execution time}
+%\label{overflow}}
+\end{figure}
 
 Using the SIMGRID simulator flexibility, we have tried to determine the 
 impact on the algorithms performance in varying the CPU power of the 
@@ -380,7 +546,7 @@ 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.
 
- \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 
@@ -408,7 +574,7 @@ best combination of the grid resources (CPU, Network, input matrix size,
 classical GMRES time. 
 
 
-The test conditions are summarized in the table below : 
+The test conditions are summarized in the table below : \\
 
 % environment
 \begin{footnotesize}
@@ -419,7 +585,7 @@ 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
- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline
+ Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline \\
  \end{tabular}
 \end{footnotesize}
 
@@ -446,27 +612,27 @@ internet.
   \centering
   \caption{Relative gain of the multisplitting algorithm compared with 
 the classical GMRES}
-  \label{tab.cluster.2x50}
+  \label{"Table 7"}
 
   \begin{mytable}{6}
     \hline
-    bw
-    & 5         & 5         & 5         & 5         & 5         & 50 \\
+    bandwidth (Mbit/s)
+    & 5         & 5         & 5         & 5         & 5 \\
     \hline
-    lat
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.02      & 0.02 \\
+    latency (ms)
+    & 20      & 20      & 20      & 20      & 20 \\
     \hline
-    power
-    & 1         & 1         & 1         & 1.5       & 1.5       & 1.5 \\
+    power (GFlops)
+    & 1         & 1         & 1         & 1.5       & 1.5 \\
     \hline
-    size
-    & 62        & 62        & 62        & 100       & 100       & 110 \\
+    size (N)
+    & 62        & 62        & 62        & 100       & 100 \\
     \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
-    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}
 
@@ -474,23 +640,23 @@ the classical GMRES}
 
   \begin{mytable}{6}
     \hline
-    bw
-    & 50        & 50        & 50        & 50        & 10        & 10 \\
+    bandwidth (Mbit/s)
+    & 50        & 50        & 50        & 50        & 50 \\
     \hline
-    lat
-    & 0.02      & 0.02      & 0.02      & 0.02      & 0.03      & 0.01 \\
+    latency (ms)
+    & 20      & 20      & 20      & 20      & 20 \\
     \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
-    size
-    & 120       & 130       & 140       & 150       & 171       & 171 \\
+    size (N)
+    & 110       & 120       & 130       & 140       & 150 \\
     \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
-    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}
@@ -505,12 +671,8 @@ CONCLUSION
 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}
 
