of asynchronous iterative algorithms. For that, we compare the behaviour of a
synchronous GMRES algorithm with an asynchronous multisplitting one with
simulations which let us easily choose some parameters. Both codes are real MPI
-codes ans simulations allow us to see when the asynchronous multisplitting algorithm can be more
+codes and simulations allow us to see when the asynchronous multisplitting algorithm can be more
efficient than the GMRES one to solve a 3D Poisson problem.
$X_{0}$ 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 involve the division of the problem
+Parallelization of such algorithms generally involves 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
convergence depends on the delay of messages. With synchronous iterations, the
number of iterations is exactly the same than in the sequential mode (if the
parallelization process does not change the algorithm). So the difficulty with
-asynchronous iteratie algorithms comes from the fact it is necessary to run the algorithm
+asynchronous iterative algorithms comes from the fact it is necessary to run the algorithm
with real data. In fact, from an execution to another the order of messages will
change and the number of iterations to reach the convergence will also change.
According to all the parameters of the platform (number of nodes, power of
-nodes, inter and intra clusrters bandwith and latency, ....) and of the
-algorithm (number of splitting with the multisplitting algorithm), the
-multisplitting code will obtain the solution more or less quickly. Or course,
+nodes, inter and intra clusrters bandwith and latency, etc.) and of the
+algorithm (number of splittings with the multisplitting algorithm), the
+multisplitting code will obtain the solution more or less quickly. Of course,
the GMRES method also depends of the same parameters. As it is difficult to have
access to many clusters, grids or supercomputers with many different network
parameters, it is interesting to be able to simulate the behaviors of
\begin{figure}[!t]
\centering
- \includegraphics[width=60mm,keepaspectratio]{clustering}
-\caption{Example of three clusters of processors interconnected by a virtual unidirectional ring network.}
+ \includegraphics[width=60mm,keepaspectratio]{clustering2}
+\caption{Example of two distant clusters of processors.}
\label{fig:4.1}
\end{figure}
& 5 & 5 & 5 & 5 & 5 \\
\hline
latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
+ & 20 & 20 & 20 & 20 & 20 \\
\hline
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 \\
& 50 & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
\hline
latency (ms)
- & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ % & 0.03 & 0.01 \\
+ & 20 & 20 & 20 & 20 & 20 \\ % & 0.03 & 0.01 \\
\hline
Power (GFlops)
& 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
\end{mytable}
\end{table}
+\RC{Du coup la latence est toujours la même, pourquoi la mettre dans la 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~\ref{tab.cluster.3x33} which shows the
%relative gains greater than 1 with a matrix size from 62 to 100 elements.
-\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
+%\CER{En accord avec RC, on a pour le moment enlevé les tableaux 2 et 3 sachant que les résultats obtenus sont limites. De même, on a enlevé aussi les deux dernières colonnes du tableau I en attendant une meilleure performance et une meilleure precision}
%\begin{table}[!t]
% \centering
% \caption{3 clusters, each with 33 nodes}
\begin{itemize}
\item 2 clusters of 50 hosts each;
\item Processor unit power: \np[GFlops]{1} or \np[GFlops]{1.5};
- \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{0.05};
- \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[$\mu$s]{20};
+ \item Intra-cluster network bandwidth: \np[Gbit/s]{1.25} and latency: \np[$\mu$s]{50};
+ \item Inter-cluster network bandwidth: \np[Mbit/s]{5} or \np[Mbit/s]{50} and latency: \np[ms]{20};
\end{itemize}
\end{itemize}
\section{Conclusion}
The simulation of the execution of parallel asynchronous iterative algorithms on large scale clusters has been presented.
In this work, we show that SIMGRID is an efficient simulation tool that allows us to
-reach the following three objectives:
+reach the following two objectives:
\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 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
-executing the algorithm in asynchronous mode.
+\item To have a flexible configurable execution platform that allows us to
+ simulate algorithms for which execution of all parts of
+ the code is necessary. Using simulations before real executions is a nice
+ solution to detect potential scalability problems.
+
+\item To test the combination of the cluster and network specifications permitting to execute an asynchronous algorithm faster than a synchronous one.
\end{enumerate}
-Our results have shown that in certain conditions, asynchronous mode is
-speeder up to \np[\%]{40} comparing to the synchronous GMRES method
+Our results have shown that with two distant clusters, the asynchronous multisplitting is faster to \np[\%]{40} compared to the synchronous GMRES method
which is not negligible for solving complex practical problems with more
and more increasing size.
tool to run efficiently an iterative parallel algorithm in asynchronous
mode in a grid architecture.
-For our futur works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters.
+In future works, we plan to extend our experimentations to larger scale platforms by increasing the number of computing cores and the number of clusters.
We will also have to increase the size of the input problem which will require the use of a more powerful simulation platform. At last, we expect to compare our simulation results to real execution results on real architectures in order to experimentally validate our study.
\section*{Acknowledgment}
