X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/f463119047e7f3b45c777e90a8702e74aba0f520..6deb73bfa86b688308eab61ad2f2cb006fd63d0f:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index f3ef835..df664fb 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 +52,10 @@
 \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}}}
 
 \usepackage{array}
 \usepackage{color, colortbl}
@@ -53,36 +66,36 @@
 \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
+\end{abstract}
 
+\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid; performance}
 
-Keywords : Algorithm distributed iterative asynchronous simulation simgrid performance
-
-\end{abstract}
+\maketitle
 
 \section{Introduction}
 
@@ -90,7 +103,70 @@ Keywords : Algorithm distributed iterative asynchronous simulation simgrid perfo
 
 \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 $A$. Algorithm~\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 of GMRES respectively.  
+
+\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 $x^k_\ell=Solve(A_{\ell\ell},c_\ell,x^{k-1}_\ell,\MIG,\TOLG)$\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}
+\label{alg:01}
+\end{algorithm}
+
+The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{Szyld92,Bru95,Bai99,bahi07}. The multisplitting methods are convergent: 
+\begin{itemize}
+\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular when the iterations are synchronous, or
+\item if $A$ is M-matrix and its splittings are regular when the iterations are asynchronous.
+\end{itemize}
+
+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 Algorithm~\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 of the two-stage algorithm. 
+ 
+
+
+
+
+
+
+
+\subsection{Simulation of two-stage methods using SimGrid framework}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Experimental, Results and Comments}
 
@@ -106,8 +182,7 @@ 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 
@@ -194,7 +269,7 @@ and our comments.
 
 \textit{3.a Executing the algorithms on various computational grid 
 architecture scaling up the input matrix size}
-
+\\
 
 % environment
 \begin{footnotesize}
@@ -209,18 +284,24 @@ architecture scaling up the input matrix size}
 
 
  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}{60mm}
+\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} 
+%\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 
@@ -237,7 +318,7 @@ 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}
 
@@ -245,11 +326,14 @@ matrix size.
 %\RCE{idem pour tous les tableaux de donnees}
 
 
-\begin{wrapfigure}{l}{45mm}
+%\begin{wrapfigure}{l}{60mm}
+\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=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). 
@@ -259,7 +343,7 @@ 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 +351,19 @@ 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}
 \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=60mm]{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 
@@ -305,11 +390,12 @@ of magnitude with a latency of 8.10$^{-6}$.
 
 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=60mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impact on execution time}
+%\label{overflow}
+\end{figure}
 
 
 
@@ -333,11 +419,12 @@ a gain of 40\% which is only around 24\% for classical GMRES.
 
 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=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 
 Nx=Ny=Nz=40 to 200 side elements that is from 40$^{3}$ = 64.000 to 
@@ -367,11 +454,12 @@ same test has been done with the grid 2x16 getting the same conclusion.
 
 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=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 
 impact on the algorithms performance in varying the CPU power of the 
@@ -446,27 +534,27 @@ 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 +562,23 @@ 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 +593,8 @@ 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}
 
