-
\documentclass[conference]{IEEEtran}
\usepackage[T1]{fontenc}
network. Parameters on the cluster's architecture are the number of machines and
the computation power of a machine. Simulations show that the asynchronous
multisplitting algorithm can solve the 3D Poisson problem approximately twice
-faster than GMRES with two distant clusters.
+faster than GMRES with two distant clusters. In this way, we present an original solution to optimize the use of a simulation
+tool to run efficiently an asynchronous iterative parallel algorithm in a grid architecture
interface, SimGrid provides bindings for the C++, Java, Lua and Ruby programming
languages. SMPI is the interface that has been used for the work exposed in
this paper. The SMPI interface implements about \np[\%]{80} of the MPI 2.0
-standard~\cite{bedaride:hal-00919507}, and supports applications written in C or
-Fortran, with little or no modifications.
+standard~\cite{bedaride+degomme+genaud+al.2013.toward}, and supports
+applications written in C or Fortran, with little or no modifications.
-Within SimGrid, the execution of a distributed application is simulated on a
-single machine. The application code is really executed, but some operations
+Within SimGrid, the execution of a distributed application is simulated by a
+single process. The application code is really executed, but some operations
like the communications are intercepted, and their running time is computed
according to the characteristics of the simulated execution platform. The
description of this target platform is given as an input for the execution, by
the mean of an XML file. It describes the properties of the platform, such as
the computing nodes with their computing power, the interconnection links with
-their bandwidth and latency, and the routing strategy. The simulated running
-time of the application is computed according to these properties.
+their bandwidth and latency, and the routing strategy. The scheduling of the
+simulated processes, as well as the simulated running time of the application is
+computed according to these properties.
To compute the durations of the operations in the simulated world, and to take
into account resource sharing (e.g. bandwidth sharing between competing
communications), SimGrid uses a fluid model. This allows to run relatively fast
simulations, while still keeping accurate
-results~\cite{bedaride:hal-00919507,tomacs13}. Moreover, depending on the
+results~\cite{bedaride+degomme+genaud+al.2013.toward,
+ velho+schnorr+casanova+al.2013.validity}. Moreover, depending on the
simulated application, SimGrid/SMPI allows to skip long lasting computations and
to only take their duration into account. When the real computations cannot be
skipped, but the results have no importance for the simulation results, there is
several simulated processes, and thus to reduce the whole memory consumption.
These two techniques can help to run simulations at a very large scale.
+The validity of simulations with SimGrid has been asserted by several studies.
+See, for example, \cite{velho+schnorr+casanova+al.2013.validity} and articles
+referenced therein for the validity of the network models. Comparisons between
+real execution of MPI applications on the one hand, and their simulation with
+SMPI on the other hand, are presented in~\cite{guermouche+renard.2010.first,
+ clauss+stillwell+genaud+al.2011.single,
+ bedaride+degomme+genaud+al.2013.toward}.
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation of the multisplitting method}
Both codes were simulated on a two clusters based network with 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
+factors have provided the results shown in Table~\ref{tab.cluster.2x50}. The problem size of the 3D Poisson problem ranges from $N=N_x = N_y = N_z = \text{62}$ to 150 elements (that is from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{150}^\text{3} =
\text{\np{3375000}}$ entries). With the asynchronous multisplitting algorithm the simulated execution time is in average 2.5 times faster than with the synchronous GMRES one.
%\AG{Expliquer comment lire les tableaux.}
\begin{table}[!t]
\centering
\caption{Relative gain of the multisplitting algorithm compared to GMRES for
- different configurations with 2 clusters, each one composed of 50 nodes.}
+ different configurations with 2 clusters, each one composed of 50 nodes. Latency = $20$ms}
\label{tab.cluster.2x50}
\begin{mytable}{5}
bandwidth (Mbit/s)
& 5 & 5 & 5 & 5 & 5 \\
\hline
- latency (ms)
- & 20 & 20 & 20 & 20 & 20 \\
- \hline
+ % latency (ms)
+ % & 20 & 20 & 20 & 20 & 20 \\
+ %\hline
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 \\
\hline
- size $(n^3)$
- & 62 & 62 & 62 & 100 & 100 \\
+ size $(N)$
+ & $62^3$ & $62^3$ & $62^3$ & $100^3$ & $100^3$ \\
\hline
Precision
& \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} \\
bandwidth (Mbit/s)
& 50 & 50 & 50 & 50 & 50 \\ % & 10 & 10 \\
\hline
- latency (ms)
- & 20 & 20 & 20 & 20 & 20 \\ % & 0.03 & 0.01 \\
- \hline
+ %latency (ms)
+ %& 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 \\
\hline
- size $(n^3)$
- & 110 & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
+ size $(N)$
+ & $110^3$ & $120^3$ & $130^3$ & $140^3$ & $150^3$ \\ % & 171 & 171 \\
\hline
Precision
& \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} \\ % & \np{E-5} & \np{E-5} \\
\end{mytable}
\end{table}
-\RC{Du coup la latence est toujours la même, pourquoi la mettre dans la 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
\begin{itemize}
\item Description of the cluster architecture matching the format <Number of
clusters> <Number of hosts in cluster1> <Number of hosts in cluster2>;
-\item Maximum number of iterations;
-\item Precisions on the residual error;
+\item Maximum numbers of outer and inner iterations;
+\item Outer and inner precisions on the residual error;
\item Matrix size $N_x$, $N_y$ and $N_z$;
-\item Matrix diagonal value: $6$ (See Equation~(\ref{eq:03}));
-\item Matrix off-diagonal value: $-1$;
+\item Matrix diagonal value: $6$ (see Equation~(\ref{eq:03}));
+\item Matrix off-diagonal values: $-1$;
\item Communication mode: asynchronous.
\end{itemize}
asynchronous multisplitting compared to GMRES with two distant clusters.
With these settings, Table~\ref{tab.cluster.2x50} shows
-that after setting the bandwidth of the inter cluster network to \np[Mbit/s]{5} and a latency in order of one hundredth of millisecond and a processor power
-of one GFlops, an efficiency of about \np[\%]{40} is
-obtained in asynchronous mode for a matrix size of 62 elements. It is noticed that the result remains
+that after setting the bandwidth of the inter cluster network to \np[Mbit/s]{5}, the latency to $20$ millisecond and the processor power
+to one GFlops, an efficiency of about \np[\%]{40} is
+obtained in asynchronous mode for a matrix size of $62^3$ elements. It is noticed that the result remains
stable even we vary the residual error precision from \np{E-5} to \np{E-9}. By
-increasing the matrix size up to 100 elements, it was necessary to increase the
+increasing the matrix size up to $100^3$ elements, it was necessary to increase the
CPU power of \np[\%]{50} to \np[GFlops]{1.5} to get the algorithm convergence and the same order of asynchronous mode efficiency. Maintaining such processor power but increasing network throughput inter cluster up to
\np[Mbit/s]{50}, the result of efficiency with a relative gain of 2.5 is obtained with
-high external precision of \np{E-11} for a matrix size from 110 to 150 side
+high external precision of \np{E-11} for a matrix size from $110^3$ to $150^3$ side
elements.
%For the 3 clusters architecture including a total of 100 hosts,
%(synchronous and asynchronous) is achieved with an inter cluster of
%\np[Mbit/s]{10} and a latency of \np[ms]{E-1}. To challenge an efficiency greater than 1.2 with a matrix %size of 100 points, it was necessary to degrade the
%inter cluster network bandwidth from 5 to \np[Mbit/s]{2}.
-\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
- Quelle est la perte de perfs en faisant ça ?}
+%\AG{Conclusion, on prend une plateforme pourrie pour avoir un bon ratio sync/async ???
+ %Quelle est la perte de perfs en faisant ça ?}
%A last attempt was made for a configuration of three clusters but more powerful
%with 200 nodes in total. The convergence with a relative gain around 1.1 was
%\CER{Définitivement, les paramètres réseaux variables ici se rapportent au réseau INTER cluster.}
\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
+In this work, we show that SimGrid is an efficient simulation tool that allows us to
reach the following two objectives:
\begin{enumerate}
\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 with two distant clusters, the asynchronous multisplitting is faster to \np[\%]{40} compared to the synchronous GMRES method
+Our results have shown that with two distant clusters, the asynchronous multisplitting method 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.
mode in a grid architecture.
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.
+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 better experimentally validate our study. Finally, we also plan to study other problems with the multisplitting method and other asynchronous iterative methods.
\section*{Acknowledgment}
