\usepackage[textsize=footnotesize]{todonotes}
\newcommand{\AG}[2][inline]{%
\todo[color=green!50,#1]{\sffamily\textbf{AG:} #2}\xspace}
+\newcommand{\DL}[2][inline]{%
+ \todo[color=yellow!50,#1]{\sffamily\textbf{DL:} #2}\xspace}
+\newcommand{\LZK}[2][inline]{%
+ \todo[color=blue!10,#1]{\sffamily\textbf{LZK:} #2}\xspace}
+\newcommand{\RC}[2][inline]{%
+ \todo[color=red!10,#1]{\sffamily\textbf{RC:} #2}\xspace}
\algnewcommand\algorithmicinput{\textbf{Input:}}
\algnewcommand\Input{\item[\algorithmicinput]}
\algnewcommand\algorithmicoutput{\textbf{Output:}}
\algnewcommand\Output{\item[\algorithmicoutput]}
+\newcommand{\MI}{\mathit{MaxIter}}
+
\begin{document}
\author{%
\IEEEauthorblockN{%
- Charles Emile Ramamonjisoa and
- David Laiymani and
- Arnaud Giersch and
- Lilia Ziane Khodja and
- Raphaël Couturier
+ Charles Emile Ramamonjisoa\IEEEauthorrefmark{1},
+ David Laiymani\IEEEauthorrefmark{1},
+ Arnaud Giersch\IEEEauthorrefmark{1},
+ Lilia Ziane Khodja\IEEEauthorrefmark{2} and
+ Raphaël Couturier\IEEEauthorrefmark{1}
+ }
+ \IEEEauthorblockA{\IEEEauthorrefmark{1}%
+ Femto-ST Institute -- DISC Department\\
+ Université de Franche-Comté,
+ IUT de Belfort-Montbéliard\\
+ 19 avenue du Maréchal Juin, BP 527, 90016 Belfort cedex, France\\
+ Email: \email{{charles.ramamonjisoa,david.laiymani,arnaud.giersch,raphael.couturier}@univ-fcomte.fr}
}
- \IEEEauthorblockA{%
- Femto-ST Institute - DISC Department\\
- Université de Franche-Comté\\
- Belfort\\
- Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
+ \IEEEauthorblockA{\IEEEauthorrefmark{2}%
+ Inria Bordeaux Sud-Ouest\\
+ 200 avenue de la Vieille Tour, 33405 Talence cedex, France \\
+ Email: \email{lilia.ziane@inria.fr}
}
}
\maketitle
-\AG{Ordre des autheurs pas définitif}
+\RC{Ordre des autheurs pas définitif.}
\begin{abstract}
-The abstract goes here.
+ABSTRACT
+
+In recent years, the scalability of large-scale implementation in a
+distributed environment of algorithms becoming more and more complex has
+always been hampered by the limits of physical computing resources
+capacity. One solution is to run the program in a virtual environment
+simulating a real interconnected computers architecture. The results are
+convincing and useful solutions are obtained with far fewer resources
+than in a real platform. However, challenges remain for the convergence
+and efficiency of a class of algorithms that concern us here, namely
+numerical parallel iterative algorithms executed in asynchronous mode,
+especially in a large scale level. Actually, such algorithm requires a
+balance and a compromise between computation and communication time
+during the execution. Two important factors determine the success of the
+experimentation: the convergence of the iterative algorithm on a large
+scale and the execution time reduction in asynchronous mode. Once again,
+from the current work, a simulated environment like Simgrid provides
+accurate results which are difficult or even impossible to obtain in a
+physical platform by exploiting the flexibility of the simulator on the
+computing units clusters and the network structure design. Our
+experimental outputs showed a saving of up to \np[\%]{40} for the algorithm
+execution time in asynchronous mode compared to the synchronous one with
+a residual precision up to \np{E-11}. Such successful results open
+perspectives on experimentations for running the algorithm on a
+simulated large scale growing environment and with larger problem size.
+
+% no keywords for IEEE conferences
+% Keywords: Algorithm distributed iterative asynchronous simulation simgrid
\end{abstract}
\section{Introduction}
\section{The asynchronous iteration model}
-Décrire le modèle asynchrone. Je m'en charge (DL)
+\DL{Décrire le modèle asynchrone. Je m'en charge}
\section{SimGrid}
-Décrire SimGrid (Arnaud)
-
-
-
-
-
+\AG{Décrire SimGrid~\cite{casanova+legrand+quinson.2008.simgrid} (Arnaud)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
%Décrire le problème (algo) traité ainsi que le processus d'adaptation à SimGrid.
-Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $y$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi partitioning to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping
+Let $Ax=b$ be a large sparse system of $n$ linear equations in $\mathbb{R}$, where $A$ is a sparse square and nonsingular matrix, $x$ is the solution vector and $b$ is the right-hand side vector. We use a multisplitting method based on the block Jacobi splitting to solve this linear system on a large scale platform composed of $L$ clusters of processors. In this case, we apply a row-by-row splitting without overlapping
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\end{array} \right)
=
\left(\begin{array}{c}
-Y_1 \\
+B_1 \\
\vdots\\
-Y_L
+B_L
\end{array} \right)\]
-in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,i\in\{1,\ldots,L\}$ $A_{li}$ is a rectangular block of $A$ of size $n_l\times n_i$, $X_l$ and $Y_l$ are sub-vectors of $x$ and $y$, respectively, each of size $n_l$ and $\sum_{l} n_l=\sum_{i} n_i=n$.
+in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$ are assigned to one cluster, where for all $l,m\in\{1,\ldots,L\}$ $A_{lm}$ is a rectangular block of $A$ of size $n_l\times n_m$, $X_l$ and $B_l$ are sub-vectors of $x$ and $b$, respectively, of size $n_l$ each and $\sum_{l} n_l=\sum_{m} n_m=n$.
-The multisplitting method proceeds by iteration to solve in parallel the linear system by $L$ clusters of processors, in such a way each sub-system
+The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
\begin{equation}
\left\{
\begin{array}{l}
A_{ll}X_l = Y_l \mbox{,~such that}\\
-Y_l = B_l - \displaystyle\sum_{i=1,i\neq l}^{L}A_{li}X_i,
+Y_l = B_l - \displaystyle\sum_{\substack{m=1\\ m\neq l}}^{L}A_{lm}X_m
\end{array}
\right.
\label{eq:4.1}
\end{equation}
-is solved independently by a cluster and communication are required to update the right-hand side sub-vectors $Y_l$, such that the sub-vectors $X_i$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
+is solved independently by a cluster and communications are required to update the right-hand side sub-vector $Y_l$, such that the sub-vectors $X_m$ represent the data dependencies between the clusters. As each sub-system (\ref{eq:4.1}) is solved in parallel by a cluster of processors, our multisplitting method uses an iterative method as an inner solver which is easier to parallelize and more scalable than a direct method. In this work, we use the parallel algorithm of GMRES method~\cite{ref1} which is one of the most used iterative method by many researchers.
\begin{algorithm}
-\caption{A multisplitting solver with inner iteration GMRES method}
+\caption{A multisplitting solver with GMRES method}
\begin{algorithmic}[1]
-\Input $A_l$ (local sparse matrix), $B_l$ (local right-hand side), $x^0$ (initial guess)
-\Output $X_l$ (local solution vector)\vspace{0.2cm}
-\State Load $A_l$, $B_l$, $x^0$
-\State Initialize the shared vector $\hat{x}=x^0$
-\For {$k=1,2,3,\ldots$ until the global convergence}
-\State $x^0=\hat{x}$
-\State Inner iteration solver: \Call{InnerSolver}{$x^0$, $k$}
-\State Exchange the local solution ${X}_l^k$ with the neighboring clusters and copy the shared vector elements in $\hat{x}$
+\Input $A_l$ (sparse sub-matrix), $B_l$ (right-hand side sub-vector)
+\Output $X_l$ (solution sub-vector)\vspace{0.2cm}
+\State Load $A_l$, $B_l$
+\State Set the initial guess $x^0$
+\For {$k=0,1,2,\ldots$ until the global convergence}
+\State Restart outer iteration with $x^0=x^k$
+\State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
+\State Send shared elements of $X_l^{k+1}$ to neighboring clusters
+\State Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
\EndFor
\Statex
\Function {InnerSolver}{$x^0$, $k$}
-\State Compute the local right-hand side: $Y_l = B_l - \sum^L_{i=1,i\neq l}A_{li}X_i^0$
-\State Solving the local splitting $A_{ll}X_l^k=Y_l$ using the parallel GMRES method, such that $X_l^0$ is the local initial guess
+\State Compute local right-hand side $Y_l$: \[Y_l = B_l - \sum\nolimits^L_{\substack{m=1 \\m\neq l}}A_{lm}X_m^0\]
+\State Solving sub-system $A_{ll}X_l^k=Y_l$ with the parallel GMRES method
\State \Return $X_l^k$
\EndFunction
\end{algorithmic}
\label{algo:01}
\end{algorithm}
+
+Algorithm~\ref{algo:01} shows the main key points of the multisplitting method to solve a large sparse linear system. This algorithm is based on an outer-inner iteration method where the parallel synchronous GMRES method is used to solve the inner iteration. It is executed in parallel by each cluster of processors. For all $l,m\in\{1,\ldots,L\}$, the matrices and vectors with the subscript $l$ represent the local data for cluster $l$, while $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with neighboring clusters. At every outer iteration $k$, asynchronous communications are performed between processors of the local cluster and those of distant clusters (lines $6$ and $7$ in Algorithm~\ref{algo:01}). The shared vector elements of the solution $x$ are exchanged by message passing using MPI non-blocking communication routines.
+
+\begin{figure}
+\centering
+ \includegraphics[width=60mm,keepaspectratio]{clustering}
+\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
+\label{fig:4.1}
+\end{figure}
+
+The global convergence of the asynchronous multisplitting solver is detected when the clusters of processors have all converged locally. We implemented the global convergence detection process as follows. On each cluster a master processor is designated (for example the processor with rank $1$) and masters of all clusters are interconnected by a virtual unidirectional ring network (see Figure~\ref{fig:4.1}). During the resolution, a Boolean token circulates around the virtual ring from a master processor to another until the global convergence is achieved. So starting from the cluster with rank $1$, each master processor $i$ sets the token to {\it True} if the local convergence is achieved or to {\it False} otherwise, and sends it to master processor $i+1$. Finally, the global convergence is detected when the master of cluster $1$ receives from the master of cluster $L$ a token set to {\it True}. In this case, the master of cluster $1$ broadcasts a stop message to masters of other clusters. In this work, the local convergence on each cluster $l$ is detected when the following condition is satisfied
+\[(k\leq \MI) \mbox{~or~} (\|X_l^k - X_l^{k+1}\|_{\infty}\leq\epsilon)\]
+where $\MI$ is the maximum number of outer iterations and $\epsilon$ is the tolerance threshold of the error computed between two successive local solution $X_l^k$ and $X_l^{k+1}$.
+
+\LZK{Description du processus d'adaptation de l'algo multisplitting à SimGrid}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\centering
\caption{2 clusters X 50 nodes}
\label{tab.cluster.2x50}
- \AG{Les images manquent dans le dépôt Git. Si ce sont vraiment des tableaux, utiliser un format vectoriel (eps ou pdf), et surtout pas de jpeg!}
+ \AG{Ces tableaux (\ref{tab.cluster.2x50}, \ref{tab.cluster.3x33} et
+ \ref{tab.cluster.3x67}) sont affreux. Utiliser un format vectoriel (eps ou
+ pdf) ou, mieux, les réécrire en \LaTeX{}. Réécrire les légendes proprement
+ également (\texttt{\textbackslash{}times} au lieu de \texttt{X} par ex.)}
\includegraphics[width=209pt]{img1.jpg}
\end{table}
\centering
\caption{3 clusters X 33 nodes}
\label{tab.cluster.3x33}
- \AG{Le fichier manque.}
+ \AG{Refaire le tableau.}
\includegraphics[width=209pt]{img2.jpg}
\end{table}
\centering
\caption{3 clusters X 67 nodes}
\label{tab.cluster.3x67}
- \AG{Le fichier manque.}
+ \AG{Refaire le tableau.}
% \includegraphics[width=160pt]{img3.jpg}
\includegraphics[scale=0.5]{img3.jpg}
\end{table}
that it was difficult to have a combination which gives an efficiency of
asynchronous below \np[\%]{80}. Indeed, for a matrix size of 62 elements, equality
between the performance of the two modes (synchronous and asynchronous) is
-achieved with an inter cluster of \np[Mbits/s]{10} and a latency of \np{E-1} ms. To
+achieved with an inter cluster of \np[Mbits/s]{10} and a latency of \np[ms]{E-1}. To
challenge an efficiency by \np[\%]{78} with a matrix size of 100 points, it was
necessary to degrade the inter cluster network bandwidth from 5 to 2 Mbit/s.
-A last attempt was made for a configuration of three clusters but more power
+A last attempt was made for a configuration of three clusters but more powerful
with 200 nodes in total. The convergence with a speedup of \np[\%]{90} was obtained
with a bandwidth of \np[Mbits/s]{1} as shown in Table~\ref{tab.cluster.3x67}.
\section{Conclusion}
+CONCLUSION
+
+The experimental results on executing a parallel iterative algorithm in
+asynchronous mode on an environment simulating a large scale of virtual
+computers organized with interconnected clusters have been presented.
+Our work has demonstrated that using such a simulation tool allow us to
+reach the following three objectives:
+
+\newcounter{numberedCntD}
+\begin{enumerate}
+\item To have a flexible configurable execution platform resolving the
+hard exercise to access to very limited but so solicited physical
+resources;
+\item to ensure the algorithm convergence with a raisonnable time and
+iteration number ;
+\item and finally and more importantly, to find the correct combination
+of the cluster and network specifications permitting to save time in
+executing the algorithm in asynchronous mode.
+\setcounter{numberedCntD}{\theenumi}
+\end{enumerate}
+Our results have shown that in certain conditions, asynchronous mode is
+speeder up to \np[\%]{40} than executing the algorithm in synchronous mode
+which is not negligible for solving complex practical problems with more
+and more increasing size.
+
+ Several studies have already addressed the performance execution time of
+this class of algorithm. The work presented in this paper has
+demonstrated an original solution to optimize the use of a simulation
+tool to run efficiently an iterative parallel algorithm in asynchronous
+mode in a grid architecture.
\section*{Acknowledgment}
