invited to read~\cite{BT89,bahi07}.
Before using an asynchronous iterative method, the convergence must be
-studied. Otherwise, there is no garantee that the application will reach the convergence. An
+studied. Otherwise, there is no guarantee that the application will reach the convergence. An
algorithm that supports both the synchronous or the asynchronous iteration model
requires very few modifications to be able to be executed in both variants. In
practice, only the communications management and the convergence detection are different. In
\section{SimGrid}
\label{sec:simgrid}
-<<<<<<< HEAD
-In the scope of this paper, we have chosen the SimGrid toolkit~\cite{SimGrid,casanova+giersch+legrand+al.2014.versatile} to simulate the behavior of parallel iterative linear solvers on different computational grid configurations. Contrary to most simulators which remain very application-oriented, the SimGrid framework is designed to study the behavior of many large-scale distributed computing platforms as Grids, Peer-to-Peer systems, Clouds or High Performance Computation systems. It is still actively developed by the scientific community and distributed as an open source software.
-
-SimGrid provides four user interfaces which can be convenient for different distributed applications~\cite{casanova+legrand+quinson.2008.simgrid}. In this paper we are interested in the SMPI user interface (Simulator MPI) which implements about \np[\%]{80} of the MPI 2.0 standard and allows minor modifications of the initial code~\cite{bedaride+degomme+genaud+al.2013.toward} (see Section~\ref{sec:04.02}). SMPI enables the direct simulation of the execution, as in real life, of an unmodified MPI distributed application, and gets accurate results with the detailed resources consumption.
-
-The SimGrid simulator uses at least three XML input files describing the computational grid resources: the number of clusters in the grid, the number of processors/cores in each cluster, the detailed description of the intra and inter networks and the list of the hosts in each cluster (see the details in Section~\ref{sec:expe}). SimGrid uses a fluid model to simulate the program execution. It allows several simulation modes which produce accurate results~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}. For instance, the "in vivo" mode really executes the computation but "intercepts" the communications (the execution time is then evaluated according to the parameters of the simulated platform). It is also possible for SimGrid/SMPI to only keep the duration of large computations by skipping them. Moreover the application can be run in an "in vitro" mode by sharing some structures between the simulated processes and thus allowing the use of very large-scale data.
-
-The choice of SimGrid/SMPI as a simulator tool in this study has been emphasized by the results obtained by several studies to validate, in real environments, the behavior of different network models simulated in SimGrid~\cite{velho+schnorr+casanova+al.2013.validity}. Other studies underline the comparison between the real MPI application executions and the SimGrid/SMPI ones~\cite{guermouche+renard.2010.first,clauss+stillwell+genaud+al.2011.single,bedaride+degomme+genaud+al.2013.toward}. These works show the accuracy of SimGrid simulations compared to the executions on real physical architectures.
-
-
-
-
-
-
-
-
-
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-=======
->>>>>>> e2082ae796c90fe550e1ad8e44f43f9d5bfedf95
In the scope of this paper, we have chosen the SimGrid
toolkit~\cite{SimGrid,casanova+legrand+quinson.2008.simgrid,casanova+giersch+legrand+al.2014.versatile}
these resources along the program execution. This model produces accurate
results while still running relatively
fast~\cite{bedaride+degomme+genaud+al.2013.toward,velho+schnorr+casanova+al.2013.validity}.
-During the simulation, the computation is really executed, but the commuications
+During the simulation, the computations are really executed, but the communications
are intercepted and their execution time evaluated according to the parameters
of the simulated platform. It is also possible for SimGrid/SMPI to only keep the
duration of large computations by skipping them. Moreover, when applicable, the
\paragraph{Simulation parameters for SimGrid}
\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
-in which several clusters are geographically distant, so there are intra and
+in which several clusters are geographically distant, so that there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
have been chosen for the study in this paper. \\
\textbf{Step 2}: Collect the software materials needed for the experimentation.
-In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+In our case, we have two variants for the resolution of the
+3D-Poisson problem: (1) using the classical GMRES; (2) using the multisplitting
method. In addition, the SimGrid simulator has been chosen to simulate the
behaviors of the distributed applications. SimGrid is running in a virtual
machine on a simple laptop. \\
represents the number of clusters in the grid and the second number represents
the number of hosts (processors/cores) in each cluster. \\
-\textbf{Step 5}: Conduct an extensive and comprehensive testings
+\textbf{Step 5}: Conduct extensive and comprehensive testings
within these configurations by varying the key parameters, especially
the CPU power capacity, the network parameters and also the size of the
input data. \\
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
-\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
- capacity" of the network is defined as the maximum of data that can transit
+\item the network bandwidth ($bw$ in Gbits/s) also known as "the data-carrying
+ capacity" of the network is defined as the maximum amount of data that can transit
from one point to another in a unit of time.
\item the network latency ($lat$ in microseconds) defined as the delay from the
- start time to send a simple data from a source to a destination.
+ starting time to send a simple data from a source to a destination.
\end{enumerate}
-Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
+Among the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
and between distant clusters. This parameter is application dependent.
- In a grid environment, it is common to distinguish, on one hand, the
+ In a grid environment, it is common to distinguish, on the one hand, the
\textit{intra-network} which refers to the links between nodes within a
cluster and on the other hand, the \textit{inter-network} which is the
backbone link between clusters. In practice, these two networks have
different speeds. The intra-network generally works like a high speed
- local network with a high bandwidth and very low latency. In opposite, the
- inter-network connects clusters sometime via heterogeneous networks components
- through internet with a lower speed. The network between distant clusters
+ local network with a high bandwidth and very low latency. On the contrary, the
+ inter-network connects clusters sometimes via heterogeneous networks components the through internet with a lower speed. The network between distant clusters
might be a bottleneck for the global performance of the application.
\subsection{Comparison between GMRES and two-stage multisplitting algorithms in
synchronous mode}
In the scope of this paper, our first objective is to analyze
-when the synchronous Krylov two-stage method has better performance than the
-classical GMRES method. With a synchronous iterative method, better performance
-means a smaller number of iterations and execution time before reaching the
+when the synchronous Krylov two-stage method has better performances than the
+classical GMRES method. With a synchronous iterative method, better performances
+mean a smaller number of iterations and execution time before reaching the
convergence.
Table~\ref{tab:01} summarizes the parameters used in the different simulations:
the grid architectures (i.e. the number of clusters and the number of nodes per
cluster), the network of inter-clusters backbone links and the matrix sizes of
the 3D Poisson problem. However, for all simulations we fix the network
-parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency
+parameters of the intra-clusters links: the bandwidth $bw$=10Gbit/s and the latency
$lat=8\mu$s. In what follows, we will present the test conditions, the output
results and our comments.
\begin{tabular}{ll}
\hline
Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
- & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbit/s, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbit/s, $lat=50\mu$s \\
\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
& $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
\end{tabular}
\end{figure}
\subsubsection{Network latency impacts on performances\\}
-Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbit/s to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
\begin{figure}[ht]
\centering
Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
solve a 3D Poisson problem of size $150^3$. The results of increasing the
-network bandwidth from $1$Gbs to $10$Gbs show the performances improvement for
+network bandwidth from $1$Gbit/s to $10$Gbit/s show the performances improvement for
both algorithms by reducing the execution times. However, the Krylov two-stage
algorithm presents a better performance gain in the considered bandwidth
interval with a gain of $40\%$ compared to only about $24\%$ for the classical
%%% fill-column: 80
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+
+% LocalWords: Ramamonjisoa Ziane Khodja Laiymani Raphaël Arnaud Giersch Femto
+% LocalWords: Franche Comté Belfort GMRES multisplitting SimGrid Krylov SMPI
+% LocalWords: MPI
