\newcommand{\MI}{\mathit{MaxIter}}
-\usepackage{array}
-\usepackage{color, colortbl}
-\newcolumntype{M}[1]{>{\centering\arraybackslash}m{#1}}
-\newcolumntype{Z}[1]{>{\raggedleft}m{#1}}
-
\begin{document}
\title{Simulation of Asynchronous Iterative Numerical Algorithms Using SimGrid}
Parallelization of such algorithms generally involved the division of the problem 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 parallel computations can be performed
-either in \emph{synchronous} communication mode where a new iteration begin only when all nodes communications are
-completed, either \emph{asynchronous} mode where processors can continue independently without or few synchronization
-points.
-
-% DL : reprendre correction ici
-Despite the effectiveness of iterative approach, a major drawback of the method is the requirement of huge
-resources in terms of computing capacity, storage and high speed communication network. Indeed, limited physical
-resources are blocking factors for large-scale deployment of parallel algorithms.
-
-In recent years, the use of a simulation environment to execute parallel iterative algorithms found some interests in
-reducing the highly cost of access to computing resources: (1) for the applications development life cycle and in code
-debugging (2) and in production to get results in a reasonable execution time with a simulated infrastructure not
-accessible with physical resources. Indeed, the launch of distributed iterative asynchronous algorithms to solve a
-given problem on a large-scale simulated environment challenges to find optimal configurations giving the best results
-with a lowest residual error and in the best of execution time. According our knowledge, no testing of large-scale
-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 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 permitted to show the
-effectiveness of asynchronous mode algorithm by 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 demonstrated not only
-the algorithm convergence within a reasonable time compared with the physical environment performance, but also a time
-saving of up to \np[\%]{40} in asynchronous mode.
+iteration starts until the approximate solution is reached. These parallel computations can be performed either in
+\emph{synchronous} communication mode where a new iteration begin only when all nodes communications are completed,
+either \emph{asynchronous} mode where processors can continue independently without or few synchronization points. For
+instance in the \textit{Asynchronous Iterations - Asynchronous Communications (AIAC)} model \cite{bcvc06:ij}, local
+computations do not need to wait for required data. Processors can then perform their iterations with the data present
+at that time. Even if the number of iterations required before the convergence is generally greater than for the
+synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
+synchronizations especially in a grid computing context (see \cite{bcvc06:ij} for more details).
+
+Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
+requirements. Quantifying their performance of resource allocation policies and application scheduling algorithms in
+grid computing environments under varying load, CPU power and network speeds is very costly, labor intensive and time
+consuming \cite{BuRaCa}. The case of AIAC algorithms is even more problematic since they are very sensible to the
+execution environment context. For instance, variations in the network bandwith (intra and inter- clusters), in the
+number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to
+very different execution times. In this context, it appears that the use of simulation tools to explore various platform
+scenarios and to run enormous numbers of experiments quickly can be very promising. In this way, the use of a simulation
+environment to execute parallel iterative algorithms found some interests in reducing the highly cost of access to
+computing resources: (1) for the applications development life cycle and in code debugging (2) and in production to get
+results in a reasonable execution time with a simulated infrastructure not accessible with physical resources. Indeed,
+the launch of distributed iterative asynchronous algorithms to solve a given problem on a large-scale simulated
+environment challenges to find optimal configurations giving the best results with a lowest residual error and in the
+best of execution time.
+
+To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
+paper is to give a first approach of the simulation of AIAC algorithms using the SimGrid toolkit \cite{SimGrid}. 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). SimGrid 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. The simulated results we obtained are in line with real results exposed in ?? In addition, it has been
+permitted to show the effectiveness of asynchronous mode algorithm by 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
+demonstrated not only the algorithm convergence within a reasonable time compared with the physical environment
+performance, but also a time 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 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 (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.
+iterative asynchronous model. Then, the simulation framework SimGrid is 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 (Simulated MPI) is detailed in the next section. At last, the experiments results
+carried out will be presented before some concluding remarks and future works.
\section{Motivations and scientific context}
algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
in a grid computing context.
-\begin{figure}[htbp]
+\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
\caption{The Asynchronous Iterations - Asynchronous Communications model }
\end{equation}
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{figure}
+\begin{figure}[!t]
%%% IEEE instructions forbid to use an algorithm environment here, use figure
%%% instead
\begin{algorithmic}[1]
elements of the solution $x$ are exchanged by message passing using MPI
non-blocking communication routines.
-\begin{figure}
+\begin{figure}[!t]
\centering
\includegraphics[width=60mm,keepaspectratio]{clustering}
\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
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 I with a matrix size
-ranging from $N_x = N_y = N_z = 62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to
-$171^{3} = \np{5211000}$ entries.
-
-\begin{table}[h!]
- \centering
+the above factors have providing the results shown in
+Table~\ref{tab.cluster.2x50} with a matrix size ranging from $N_x = N_y = N_z =
+62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
+\np{5211000}$ entries.
- \tiny
-
-\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|M{0.25cm}|}
- \hline
- \bf bw & 5 &5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 & 10 & 10\\
- \hline
- \bf lat & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01\\
- \hline
- \bf power & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5\\ \hline \bf size & 62 & 62 & 62 & 100 & 100 & 110 & 120& 130 & 140 & 150 & 171 & 171\\ \hline
- \bf Prec/Eprec & 10$^{-5}$ & 10$^{-8}$ & 10$^{-9}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-11}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline
- \bf speedup & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778\\ \hline
- \end{tabular}
- \smallskip
- \caption{2 Clusters x 50 nodes each} \label{tab1}
-\end{table}
+\begin{table}[!t]
+ \centering
+ \caption{2 clusters, each with 50 nodes}
+ \label{tab.cluster.2x50}
+ \renewcommand{\arraystretch}{1.3}
+
+ \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+ \hline
+ bw
+ & 5 & 5 & 5 & 5 & 5 & 50 \\
+ \hline
+ lat
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ \hline
+ power
+ & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 \\
+ \hline
+ size
+ & 62 & 62 & 62 & 100 & 100 & 110 \\
+ \hline
+ Prec/Eprec
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ \hline
+ speedup
+ & 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
+ \hline
+ \end{tabular}
+
+ \smallskip
+
+ \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+ \hline
+ bw
+ & 50 & 50 & 50 & 50 & 10 & 10 \\
+ \hline
+ lat
+ & 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\
+ \hline
+ power
+ & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 \\
+ \hline
+ size
+ & 120 & 130 & 140 & 150 & 171 & 171 \\
+ \hline
+ Prec/Eprec
+ & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\
+ \hline
+ speedup
+ & 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
+ \hline
+ \end{tabular}
+\end{table}
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
clusters. In the same way as above, a judicious choice of key parameters has
-permitted to get the results in Table II which shows the speedups less than 1 with
-a matrix size from 62 to 100 elements.
-
-\begin{table}[h!]
- \centering
+permitted to get the results in Table~\ref{tab.cluster.3x33} which shows the
+speedups less than 1 with a matrix size from 62 to 100 elements.
- \tiny
-
-\begin{tabular}{|Z{0.55cm}|Z{0.25cm}|Z{0.25cm}|M{0.25cm}|Z{0.25cm}|M{0.25cm}|M{0.25cm}|}
- \hline
- \bf bw & 10 &5 & 4 & 3 & 2 & 6\\ \hline
- \bf lat & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02\\
- \hline
- \bf power & 1 & 1 & 1 & 1 & 1 & 1\\ \hline
- \bf size & 62 & 100 & 100 & 100 & 100 & 171\\ \hline
- \bf Prec/Eprec & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$ & 10$^{-5}$\\ \hline
- \bf speedup & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99\\
- \hline
- \end{tabular}
- \smallskip
- \caption{3 Clusters x 33 nodes each} \label{tab2}
-\end{table}
+\begin{table}[!t]
+ \centering
+ \caption{3 clusters, each with 33 nodes}
+ \label{tab.cluster.3x33}
+ \renewcommand{\arraystretch}{1.3}
+
+ \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
+ \hline
+ bw
+ & 10 & 5 & 4 & 3 & 2 & 6 \\
+ \hline
+ lat
+ & 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ \hline
+ power
+ & 1 & 1 & 1 & 1 & 1 & 1 \\
+ \hline
+ size
+ & 62 & 100 & 100 & 100 & 100 & 171 \\
+ \hline
+ Prec/Eprec
+ & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
+ \hline
+ speedup
+ & 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
+ \hline
+ \end{tabular}
+\end{table}
In a final step, results of an execution attempt to scale up the three clustered
-configuration but increasing by two hundreds hosts has been recorded in Table III.
-
-\begin{table}[h!]
- \centering
- \tiny
-\begin{tabular}{|M{0.55cm}|M{0.25cm}|}
- \hline
- \bf bw & 1\\ \hline
- \bf lat & 0.02\\
- \hline
- \bf power & 1\\
- \hline
- \bf size & 62\\
- \hline
- \bf Prec/Eprec & 10$^{-5}$\\
- \hline
- \bf speedup & 0.9\\
- \hline
+configuration but increasing by two hundreds hosts has been recorded in
+Table~\ref{tab.cluster.3x67}.
+
+\begin{table}[!t]
+ \centering
+ \caption{3 clusters, each with 66 nodes}
+ \label{tab.cluster.3x67}
+ \renewcommand{\arraystretch}{1.3}
+
+ \begin{tabular}{|>{\bfseries}r|c|}
+ \hline
+ bw & 1 \\
+ \hline
+ lat & 0.02 \\
+ \hline
+ power & 1 \\
+ \hline
+ size & 62 \\
+ \hline
+ Prec/Eprec & \np{E-5} \\
+ \hline
+ speedup & 0.9 \\
+ \hline
\end{tabular}
- \smallskip
- \caption{3 Clusters x 66 nodes each} \label{tab3}
-\end{table}
+\end{table}
Note that the program was run with the following parameters:
\item Execution Mode: synchronous or asynchronous.
\end{itemize}
-
\paragraph*{Interpretations and comments}
After analyzing the outputs, generally, for the configuration with two or three
-clusters including one hundred hosts (Table I and II), some combinations of the
-used parameters affecting the results have given a speedup less than 1, showing
-the effectiveness of the asynchronous performance compared to the synchronous
-mode.
-
-In the case of a two clusters configuration, Table I shows that with a
-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 \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 \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 II shows
-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[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.
+clusters including one hundred hosts (Tables~\ref{tab.cluster.2x50}
+and~\ref{tab.cluster.3x33}), some combinations of the used parameters affecting
+the results have given a speedup less than 1, showing the effectiveness of the
+asynchronous performance compared to the synchronous 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 \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 \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
+\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 \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[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 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 III.
+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}
The experimental results on executing a parallel iterative algorithm in
