X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/239f7fccc1064a9fccc94e25ce91e23a297402c1..d2f6dd23c329a1dc4056aa229e08119172290acb:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index 0ec97f3..77dde18 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,3 @@
-%\documentclass[conference]{IEEEtran}
 \documentclass[times]{cpeauth}
 
 \usepackage{moreverb}
@@ -99,7 +98,29 @@ ABSTRACT
 
 \section{SimGrid}
 
-\section{Simulation of the multisplitting method}
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Two-stage splitting methods}
+\label{sec:04}
+\subsection{Multisplitting methods for sparse linear systems}
+\label{sec:04.01}
+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. The multisplitting methods solve the linear system~(\ref{eq:01}) iteratively as follows: 
+\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
+\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.
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experimental, Results and Comments}
 
@@ -226,14 +247,14 @@ 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. 
 
-%%\begin{wrapfigure}{l}{60mm}
+%\begin{wrapfigure}{l}{60mm}
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Cluster_x_Nodes_NX=150_and_NX=170.jpg}
-\caption{Cluster x Nodes NX=150 and NX=170 \label{overflow}}
+\includegraphics[width=60mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\caption{Cluster x Nodes NX=150 and NX=170} 
+%\label{overflow}}
 \end{figure}
-%%\end{wrapfigure}
-
+%\end{wrapfigure}
 
 Unless the 8x8 cluster, the time 
 execution difference between the two algorithms is important when 
@@ -260,11 +281,14 @@ matrix size.
 %\RCE{idem pour tous les tableaux de donnees}
 
 
+%\begin{wrapfigure}{l}{60mm}
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Cluster_x_Nodes_N1_x_N2.jpg}
-\caption{Cluster x Nodes N1 x N2\label{overflow}}
+\includegraphics[width=60mm]{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). 
@@ -291,8 +315,9 @@ Table 3 : Network latency impact
 
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Network_latency_impact_on_execution_time.jpg}
-\caption{Network latency impact on execution time\label{overflow}}
+\includegraphics[width=60mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impact on execution time}
+%\label{overflow}}
 \end{figure}
 
 
@@ -322,8 +347,9 @@ Table 4 : Network bandwidth impact
 
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Network_bandwith_impact_on_execution_time.jpg}
-\caption{Network bandwith impact on execution time\label{overflow}}
+\includegraphics[width=60mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impact on execution time}
+%\label{overflow}
 \end{figure}
 
 
@@ -350,8 +376,9 @@ Table 5 : Input matrix size impact
 
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{Pb_size_impact_on_execution_time.jpg}
-\caption{Pb size impact on execution time\label{overflow}}
+\includegraphics[width=60mm]{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 
@@ -384,8 +411,9 @@ Table 6 : CPU Power impact
 
 \begin{figure} [ht!]
 \centering
-\includegraphics[width=60mm]{CPU_Power_impact_on_execution_time.jpg}
-\caption{CPU Power impact on execution time\label{overflow}}
+\includegraphics[width=60mm]{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 
