X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/a0fcf29838c2b115bdb217bc2966c244cdb09649..599f8f6cf156ba20bf3ce06d2890246efdf0dc5d:/paper.tex diff --git a/paper.tex b/paper.tex index b7d5fef..26eb7d6 100644 --- a/paper.tex +++ b/paper.tex @@ -1,4 +1,3 @@ -%\documentclass[conference]{IEEEtran} \documentclass[times]{cpeauth} \usepackage{moreverb} @@ -53,7 +52,8 @@ \algnewcommand\algorithmicoutput{\textbf{Output:}} \algnewcommand\Output{\item[\algorithmicoutput]} -\newcommand{\MI}{\mathit{MaxIter}} +\newcommand{\TOLG}{\mathit{tol_{gmres}}} +\newcommand{\MIG}{\mathit{maxit_{gmres}}} \usepackage{array} \usepackage{color, colortbl} @@ -85,6 +85,8 @@ 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 +101,78 @@ 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 $A$~\cite{O'leary85,White86}, 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 cluster of processors is responsible for solving one splitting as a linear sub-system +\begin{equation} +M_\ell y_\ell = c_\ell^k,\mbox{~such that~} c_\ell^k = N_\ell x^k + b, +\label{eq:03} +\end{equation} +then the weighting matrices $E_\ell$ are used to compute the solution of the global system~(\ref{eq:01}) +\begin{equation} +x^{k+1}=\displaystyle\sum^L_{\ell=1} E_\ell y_\ell. +\label{eq:04} +\end{equation} +The convergence of the multisplitting methods, based on synchronous or asynchronous iterations, is studied by many authors for example~\cite{O'leary85,bahi97,Bai99,bahi07}. %It is dependent on the condition +%\begin{equation} +%\rho(\displaystyle\sum_{\ell=1}^L E_\ell M^{-1}_\ell N_\ell) < 1, +%\label{eq:05} +%\end{equation} +%where $\rho$ is the spectral radius of the square matrix. +The multisplitting methods are convergent: +\begin{itemize} +\item if $A^{-1}>0$ and the splittings of matrix $A$ are weak regular (i.e. $M^{-1}\geq 0$ and $M^{-1}N\geq 0$) when the iterations are synchronous, or +\item if $A$ is M-matrix and its splittings are regular (i.e. $M^{-1}\geq 0$ and $N\geq 0$) when the iterations are asynchronous. +\end{itemize} +The solutions of the different linear sub-systems~(\ref{eq:03}) arising from the multisplitting of matrix $A$ can be either computed exactly with a direct method or approximated with an iterative method. In the latter case, the multisplitting methods are called {\it inner-outer iterative methods} or {\it two-stage multisplitting methods}. This kind of methods uses two nested iterations: the outer iteration and the inner iteration (that of the iterative method). + +In this paper we are focused on two-stage multisplitting methods, in their both versions synchronous and asynchronous, where the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} is used as an inner iteration. Furthermore, 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. weighting matrices $E_\ell$ have only zero and one factors). In this case, the iteration of the multisplitting method presented by (\ref{eq:03}) and~(\ref{eq:04}) can be rewritten in the following form +\begin{equation} +A_{\ell\ell} x_\ell^{k+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:05} +\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. In each outer iteration $k$ until the convergence, each sub-system arising from the block Jacobi multisplitting +\begin{equation} +A_{\ell\ell} x_\ell = c_\ell, +\label{eq:06} +\end{equation} +is solved iteratively using GMRES method and independently from other sub-systems by a cluster of processors. The right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. Algorithm~\ref{alg:01} shows the main key points of the block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:06}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of iterations and the tolerance threshold respectively. + +\begin{algorithm}[t] +\caption{Block Jacobi two-stage 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 + \State Receive $\{x_m^k\}_{m\neq\ell}$ from neighboring clusters + \EndFor +\end{algorithmic} +\label{alg:01} +\end{algorithm} + +\subsection{Simulation of two-stage methods using SimGrid framework} + +%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%% \section{Experimental, Results and Comments} @@ -115,8 +188,7 @@ 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 +298,14 @@ 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 +332,14 @@ 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 +366,9 @@ 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 +398,9 @@ 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 +427,9 @@ 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 +462,9 @@ 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 @@ -520,12 +599,8 @@ 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}