X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/239f7fccc1064a9fccc94e25ce91e23a297402c1..6deb73bfa86b688308eab61ad2f2cb006fd63d0f:/paper.tex

diff --git a/paper.tex b/paper.tex
index 0ec97f3..df664fb 100644
--- a/paper.tex
+++ b/paper.tex
@@ -1,4 +1,3 @@
-%\documentclass[conference]{IEEEtran}
 \documentclass[times]{cpeauth}
 
 \usepackage{moreverb}
@@ -53,7 +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}
@@ -85,6 +87,8 @@
     Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
 }
 
+%% 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}
@@ -99,7 +103,70 @@ ABSTRACT
 
 \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}
 
@@ -115,8 +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 
@@ -226,14 +292,14 @@ 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 +326,14 @@ 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 +360,9 @@ 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 +392,9 @@ 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 +421,9 @@ 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 +456,9 @@ 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 
@@ -461,27 +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}
 
@@ -489,23 +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}
@@ -520,12 +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}
 
