\maketitle
-\RC{Ordre des autheurs pas définitif.}
+\RC{Ordre des auteurs pas définitif.}
\begin{abstract}
In recent years, the scalability of large-scale implementation in a
distributed environment of algorithms becoming more and more complex has
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 class of highly efficient
-parallel algorithms called \texttt{numerical iterative algorithms} executed in a distributed environment. As their name
-suggests, these algorithm solves a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
+parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
+suggests, these algorithms solve a given problem 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 residual error. Several well-known methods
demonstrate the convergence of these algorithms \cite{BT89,Bahi07}.
-Parallelization of such algorithms generally involved the division of the problem into several \emph{blocks} that will
+Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will
be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
iteration starts and until the approximate solution is reached. These parallel computations can be performed either in
-\emph{synchronous} 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
+\emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
+or in \emph{asynchronous} mode where processors can continue independently with few or no 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
requirements. Quantifying their resource allocation policies and application scheduling algorithms in
grid computing environments under varying load, CPU power and network speeds is very costly, very labor intensive and very time
consuming \cite{Calheiros:2011:CTM:1951445.1951450}. 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
+execution environment context. For instance, variations in the network bandwidth (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. Then, it appears that the use of simulation tools to explore various platform
scenarios and to run large numbers of experiments quickly can be very promising. In this way, the use of a simulation
SimGrid toolkit \cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their
performance with the synchronous mode. More precisely, we had implemented a program for solving large non-symmetric
linear system of equations by numerical method GMRES (Generalized Minimal Residual) []. We show, that with minor
-modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campain of a real AIAC
+modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC
application on different computing architectures. The simulated results we obtained are in line with real results
-exposed in ??. 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. It has been
-permitted to show With selected parameters on the network platforms (bandwidth, latency of inter cluster network) and
+exposed in ??\AG[]{??}. 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.
+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 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
+distributed architectures. The algorithm of the multisplitting 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.
As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
classified in three main classes depending on how iterations and communications are managed (for more details readers
-can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations - Synchronous Communications (SISC)} model data
+can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data
are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
-important idle times on processors are generated. The \textit{Synchronous Iterations - Asynchronous Communications
+important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications
(SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
i.e. without stopping current computations. This technique allows to partially overlap communications by computations
but unfortunately, the overlapping is only partial and important idle times remain. It is clear that, in a grid
computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
-\textit{Asynchronous Iterations - Asynchronous Communications (AIAC)} model. Here, local computations do not need to
+\textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to
wait for required data. Processors can then perform their iterations with the data present at that time. Figure
-\ref{fig:aiac} illustrates this model where the grey blocks represent the computation phases, the white spaces the idle
+\ref{fig:aiac} illustrates this model where the gray blocks represent the computation phases, the white spaces the idle
times and the arrows the communications. With this algorithmic model, the number of iterations required before the
convergence is generally greater than for the two former classes. But, and as detailed in \cite{bcvc06:ij}, AIAC
algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
\begin{figure}[!t]
\centering
\includegraphics[width=8cm]{AIAC.pdf}
- \caption{The Asynchronous Iterations - Asynchronous Communications model }
+ \caption{The Asynchronous Iterations~-- Asynchronous Communications model}
\label{fig:aiac}
\end{figure}
very promising. Several works...
In the context of AIAC algorithms, the use of simulation tools is even more relevant. Indeed, this class of applications
-is very sensible to the execution environment context. For instance, variations in the network bandwith (intra and
+is very sensible to the execution environment context. For instance, variations in the network bandwidth (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.
\section{SimGrid}
-SimGrid~\cite{casanova+legrand+quinson.2008.simgrid,SimGrid} is a simulation
-framework to sudy the behavior of large-scale distributed systems. As its name
+SimGrid~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid} is a simulation
+framework to study the behavior of large-scale distributed systems. As its name
says, it emanates from the grid computing community, but is nowadays used to
study grids, clouds, HPC or peer-to-peer systems. The early versions of SimGrid
-date from 1999, but it's still actively developped and distributed as an open
+date from 1999, but it's still actively developed and distributed as an open
source software. Today, it's one of the major generic tools in the field of
simulation for large-scale distributed systems.
We did not encounter major blocking problems when adapting the multisplitting algorithm previously described to a simulation environment like SIMGRID unless some code
debugging. Indeed, apart from the review of the program sequence for asynchronous exchanges between the six neighbors of each point in a submatrix within a cluster or
between clusters, the algorithm was executed successfully with SMPI and provided identical outputs as those obtained with direct execution under MPI. In synchronous
-mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_waitall instructions
-and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non- convergence of the algorithm. Note here that the use of SMPI
+mode, the execution of the program raised no particular issue but in asynchronous mode, the review of the sequence of MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions
+and with the addition of the primitive MPI\_Test was needed to avoid a memory fault due to an infinite loop resulting from the non-convergence of the algorithm. Note here that the use of SMPI
functions optimizer for memory footprint and CPU usage is not recommended knowing that one wants to get real results by simulation.
As mentioned, upon this adaptation, the algorithm is executed as in the real life in the simulated environment after the following minor changes. First, all declared
global variables have been moved to local variables for each subroutine. In fact, global variables generate side effects arising from the concurrent access of
-shared memory used by threads simulating each computing units in the Simgrid architecture. Second, the alignment of certain types of variables such as "long int" had
-also to be reviewed. Finally, some compilation errors on MPI\_waitall and MPI\_Finalise primitives have been fixed with the latest version of Simgrid.
+shared memory used by threads simulating each computing units in the SimGrid architecture. Second, the alignment of certain types of variables such as ``long int'' had
+also to be reviewed. Finally, some compilation errors on MPI\_Waitall and MPI\_Finalize primitives have been fixed with the latest version of SimGrid.
In total, the initial MPI program running on the simulation environment SMPI gave after a very simple adaptation the same results as those obtained in a real
environment. We have tested in synchronous mode with a simulated platform starting from a modest 2 or 3 clusters grid to a larger configuration like simulating
Grid5000 with more than 1500 hosts with 5000 cores~\cite{bolze2006grid}. Once the code debugging and adaptation were complete, the next section shows our methodology and experimental
62 \text{ to } 171$ elements or from $62^{3} = \np{238328}$ to $171^{3} =
\np{5211000}$ entries.
+% use the same column width for the following three tables
+\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
+\newenvironment{mytable}[1]{% #1: number of columns for data
+ \renewcommand{\arraystretch}{1.3}%
+ \begin{tabular}{|>{\bfseries}r%
+ |*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
+ \end{tabular}}
+
\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|}}
+ \begin{mytable}{6}
\hline
bw
& 5 & 5 & 5 & 5 & 5 & 50 \\
speedup
& 0.396 & 0.392 & 0.396 & 0.391 & 0.393 & 0.395 \\
\hline
- \end{tabular}
+ \end{mytable}
\smallskip
- \begin{tabular}{|>{\bfseries}r|*{12}{c|}}
+ \begin{mytable}{6}
\hline
bw
& 50 & 50 & 50 & 50 & 10 & 10 \\
speedup
& 0.398 & 0.388 & 0.393 & 0.394 & 0.63 & 0.778 \\
\hline
- \end{tabular}
+ \end{mytable}
\end{table}
Then we have changed the network configuration using three clusters containing
\centering
\caption{$3$ clusters, each with $33$ nodes}
\label{tab.cluster.3x33}
- \renewcommand{\arraystretch}{1.3}
- \begin{tabular}{|>{\bfseries}r|*{6}{c|}}
+ \begin{mytable}{6}
\hline
bw
& 10 & 5 & 4 & 3 & 2 & 6 \\
speedup
& 0.997 & 0.99 & 0.93 & 0.84 & 0.78 & 0.99 \\
\hline
- \end{tabular}
+ \end{mytable}
\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~\ref{tab.cluster.3x67}.
\centering
\caption{3 clusters, each with 66 nodes}
\label{tab.cluster.3x67}
- \renewcommand{\arraystretch}{1.3}
- \begin{tabular}{|>{\bfseries}r|c|}
+ \begin{mytable}{1}
\hline
bw & 1 \\
\hline
\hline
speedup & 0.9 \\
\hline
- \end{tabular}
+ \end{mytable}
\end{table}
Note that the program was run with the following parameters:
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
+that with a deterioration of inter cluster network set with \np[Mbit/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
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
+\np[Mbit/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.
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[Mbit/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.
+inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
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
+obtained with a bandwidth of \np[Mbit/s]{1} as shown in
Table~\ref{tab.cluster.3x67}.
\section{Conclusion}
\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
+\item to ensure the algorithm convergence with a reasonable 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
%%% fill-column: 80
%%% ispell-local-dictionary: "american"
%%% End:
+
+% LocalWords: Ramamonjisoa Laiymani Arnaud Giersch Ziane Khodja Raphaël Femto
+% LocalWords: Université Franche Comté IUT Montbéliard Maréchal Juin Inria Sud
+% LocalWords: Ouest Vieille Talence cedex scalability experimentations HPC MPI
+% LocalWords: Parallelization AIAC GMRES multi SMPI SISC SIAC SimDAG DAGs Lua
+% LocalWords: Fortran GFlops priori Mbit de du fcomte multisplitting scalable
+% LocalWords: SimGrid Belfort parallelize Labex ANR LABX IEEEabrv hpccBib
