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+\newcommand{\MI}{\mathit{MaxIter}}
+
\begin{document}
\author{%
\IEEEauthorblockN{%
- Raphaël Couturier,
- Arnaud Giersch,
+ Charles Emile Ramamonjisoa and
David Laiymani and
- Charles Emile Ramamonjisoa
+ Arnaud Giersch and
+ Lilia Ziane Khodja and
+ Raphaël Couturier
}
\IEEEauthorblockA{%
Femto-ST Institute - DISC Department\\
Université de Franche-Comté\\
Belfort\\
- Email: raphael.couturier@univ-fcomte.fr
+ Email: \email{{raphael.couturier,arnaud.giersch,david.laiymani,charles.ramamonjisoa}@univ-fcomte.fr}
}
}
\maketitle
+\RC{Ordre des autheurs pas définitif}
\begin{abstract}
The abstract goes here.
\end{abstract}
the field. Indeed, the increasing complexity of these requested
applications combined with a continuous increase of their sizes lead to
write distributed and parallel algorithms requiring significant hardware
-resources ( grid computing , clusters, broadband network ,etc... ) but
-also a non- negligible CPU execution time. We consider in this paper a
+resources (grid computing, clusters, broadband network, etc\dots{}) but
+also a non-negligible CPU execution time. We consider in this paper a
class of highly efficient parallel algorithms called iterative executed
in a distributed environment. As their name suggests, these algorithm
solves a given problem that might be NP- complete complex by successive
-iterations (X$_{n +1 }$= f (X$_{n}$) ) from an initial value X
-$_{0}$ to find an approximate value X* of the solution with a very low
+iterations ($X_{n +1} = f(X_{n})$) from an initial value $X_{0}$ to find
+an approximate value $X^*$ of the solution with a very low
residual error. Several well-known methods demonstrate the convergence
of these algorithms. Generally, to reduce the complexity and the
-execution time, the problem is divided into several "pieces" that will
+execution time, the problem is divided into several \emph{pieces} that will
be solved in parallel on multiple processing units. The latter will
communicate each intermediate results before a new iteration starts
until the approximate solution is reached. These distributed parallel
-computations can be performed either in "synchronous" communication mode
+computations can be performed either in \emph{synchronous} communication mode
where a new iteration begin only when all nodes communications are
-completed, either "asynchronous" mode where processors can continue
+completed, either \emph{asynchronous} mode where processors can continue
independently without or few synchronization points. Despite the
effectiveness of iterative approach, a major drawback of the method is
the requirement of huge resources in terms of computing capacity,
simulation of the class of algorithm solving to achieve real results has
been undertaken to date. We had in the scope of this work implemented a
program for solving large non-symmetric linear system of equations by
-numerical method GMRES (Generalized Minimal Residual ) in the simulation
-environment Simgrid . The simulated platform had allowed us to launch
+numerical method GMRES (Generalized Minimal Residual) in the simulation
+environment SimGrid. The simulated platform had allowed us to launch
the application from a modest computing infrastructure by simulating
different distributed architectures composed by clusters nodes
interconnected by variable speed networks. In addition, it has been
comparing its performance with the synchronous mode time. With selected
parameters on the network platforms (bandwidth, latency of inter cluster
network) and on the clusters architecture (number, capacity calculation
-power) in the simulated environment , the experimental results have
+power) in the simulated environment, the experimental results have
demonstrated not only the algorithm convergence within a reasonable time
compared with the physical environment performance, but also a time
-saving of up to 40 \% in asynchronous mode.
+saving of up to \np[\%]{40} in asynchronous mode.
This article is structured as follows: after this introduction, the next
section will give a brief description of iterative asynchronous model.
-Then, the simulation framework SIMGRID will be presented with the
+Then, the simulation framework SimGrid will be presented with the
settings to create various distributed architectures. The algorithm of
the multi -splitting method used by GMRES written with MPI primitives
-and its adaptation to Simgrid with SMPI (Simulation MPI ) will be in the
-next section . At last, the experiments results carried out will be
+and its adaptation to SimGrid with SMPI (Simulated MPI) will be in the
+next section. At last, the experiments results carried out will be
presented before the conclusion which we will announce the opening of
our future work after the results.
\section{SimGrid}
-Décrire SimGrid (Arnaud)
+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 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
\[
\left(\begin{array}{ccc}
A_{11} & \cdots & A_{1L} \\
\end{array} \right)
=
\left(\begin{array}{c}
-Y_1 \\
+B_1 \\
\vdots\\
-Y_L
+B_L
\end{array} \right)\]
-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, each of size $n_l$ 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 communication 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 Initialize the solution vector $x^0$
+\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 such that the parallel 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 $1$ receive from the master of cluster $L$ a token set to {\it True}. In this case, the master of cluster $1$ sends 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 $X_l^{k+1}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
network configuration where the synchronous mode will take advantage on the rapid
exchange of information on such high-speed links. Thus, the methodology adopted
was to launch the application on clustered network. In this last configuration,
-degrading the inter-cluster network performance will "penalize" the synchronous
+degrading the inter-cluster network performance will \emph{penalize} the synchronous
mode allowing to get a speedup lower than 1. This action simulates the case of
clusters linked with long distance network like Internet.
As a first step, the algorithm was run on a network consisting of two clusters
containing fifty hosts each, totaling one hundred hosts. Various combinations of
the above factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a matrix size
-ranging from Nx = Ny = Nz = 62 to 171 elements or from 62$^{3}$ = 238328 to
-171$^{3}$ = 5,211,000 entries.
+ranging from Nx = Ny = Nz = 62 to 171 elements or from $62^{3} = \np{238328}$ to
+$171^{3} = \np{5211000}$ entries.
Then we have changed the network configuration using three clusters containing
respectively 33, 33 and 34 hosts, or again by on hundred hosts for all the
\paragraph*{SMPI parameters}
\begin{itemize}
- \item HOSTFILE : Hosts file description.
+ \item HOSTFILE: Hosts file description.
\item PLATFORM: file description of the platform architecture : clusters (CPU power,
-... ) , intra cluster network description, inter cluster network (bandwidth bw ,
-lat latency , ... ).
+\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
+lat latency, \dots{}).
\end{itemize}
\item Description of the cluster architecture;
\item Maximum number of internal and external iterations;
\item Internal and external precisions;
- \item Matrix size NX , NY and NZ;
+ \item Matrix size NX, NY and NZ;
\item Matrix diagonal value = 6.0;
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
\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!}
\includegraphics[width=209pt]{img1.jpg}
\end{table}
\centering
\caption{3 clusters X 33 nodes}
\label{tab.cluster.3x33}
+ \AG{Le fichier manque.}
\includegraphics[width=209pt]{img2.jpg}
\end{table}
\centering
\caption{3 clusters X 67 nodes}
\label{tab.cluster.3x67}
+ \AG{Le fichier manque.}
% \includegraphics[width=160pt]{img3.jpg}
\includegraphics[scale=0.5]{img3.jpg}
\end{table}
mode.
In the case of a two clusters configuration, Table~\ref{tab.cluster.2x50} shows that with a
-deterioration of inter cluster network set with 5 Mbits/s of bandwidth, a latency
+deterioration of inter cluster network set with \np[Mbits/s]{5} of bandwidth, a latency
in order of a hundredth of a millisecond and a system power of one GFlops, an
-efficiency of about 40\% in asynchronous mode is obtained for a matrix size of 62
-elements . It is noticed that the result remains stable even if we vary the
-external precision from E -05 to E-09. By increasing the problem size up to 100
-elements, it was necessary to increase the CPU power of 50 \% to 1.5 GFlops for a
+efficiency of about \np[\%]{40} in asynchronous mode is obtained for a matrix size of 62
+elements. It is noticed that the result remains stable even if we vary the
+external precision from \np{E-5} to \np{E-9}. By increasing the problem size up to 100
+elements, it was necessary to increase the CPU power of \np[\%]{50} to \np[GFlops]{1.5} for a
convergence of the algorithm with the same order of asynchronous mode efficiency.
Maintaining such a system power but this time, increasing network throughput
-inter cluster up to 50 Mbits /s, the result of efficiency of about 40\% is
-obtained with high external precision of E-11 for a matrix size from 110 to 150
-side elements .
+inter cluster up to \np[Mbits/s]{50}, the result of efficiency of about \np[\%]{40} is
+obtained with high external precision of \np{E-11} for a matrix size from 110 to 150
+side elements.
For the 3 clusters architecture including a total of 100 hosts, Table~\ref{tab.cluster.3x33} shows
that it was difficult to have a combination which gives an efficiency of
-asynchronous below 80 \%. Indeed, for a matrix size of 62 elements, equality
+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 10 Mbits/s and a latency of E- 01 ms. To
-challenge an efficiency by 78\% with a matrix size of 100 points, it was
+achieved with an inter cluster of \np[Mbits/s]{10} and a latency of \np{E-1} ms. 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
-with 200 nodes in total. The convergence with a speedup of 90 \% was obtained
-with a bandwidth of 1 Mbits/s as shown in Table~\ref{tab.cluster.3x67}.
+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}
\section*{Acknowledgment}
-The authors would like to thank...
+The authors would like to thank\dots{}
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