%% execution time.
%% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
-
The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
-In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the Multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the GMRES algorithm.
+In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
+
\end{abstract}
%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
from its neighbors. We say that the iteration computation follows a
\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
iteration without having to wait for the data dependencies coming from its
-neighbors. Both communication and computations are \textit{asynchronous}
+neighbors. Both communications and computations are \textit{asynchronous}
inducing that there is no more idle time, due to synchronizations, between two
iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
-that we detail in section~\ref{sec:asynchro} but even if the number of
+that we detail in Section~\ref{sec:asynchro} but even if the number of
iterations required to converge is generally greater than for the synchronous
case, it appears that the asynchronous iterative scheme can significantly
reduce overall execution times by suppressing idle times due to
challenging and labor intensive~\cite{Calheiros:2011:CTM:1951445.1951450}. This
problematic is even more difficult for the asynchronous scheme where a small
parameter variation of the execution platform and of the application data can
-lead to very different numbers of iterations to reach the converge and so to
+lead to very different numbers of iterations to reach the convergence and so to
very different execution times. In this challenging context we think that the
use of a simulation tool can greatly leverage the possibility of testing various
platform scenarios.
The {\bf main contribution of this paper} is to show that the use of a
simulation tool (i.e. the SimGrid toolkit~\cite{SimGrid}) in the context of real
parallel applications (i.e. large linear system solvers) can help developers to
-better tune their application for a given multi-core architecture. To show the
+better tune their applications for a given multi-core architecture. To show the
validity of this approach we first compare the simulated execution of the Krylov
-multisplitting algorithm with the GMRES (Generalized Minimal Residual)
+multisplitting algorithm with the GMRES (Generalized Minimal RESidual)
solver~\cite{saad86} in synchronous mode. The simulation results allow us to
-determine which method to choose given a specified multi-core architecture.
+determine which method to choose for a given multi-core architecture.
Moreover the obtained results on different simulated multi-core architectures
confirm the real results previously obtained on non simulated architectures.
More precisely the simulated results are in accordance (i.e. with the same order
of magnitude) with the works presented in~\cite{couturier15}, which show that
-the synchronous multisplitting method is more efficient than GMRES for large
+the synchronous Krylov multisplitting method is more efficient than GMRES for large
scale clusters. Simulated results also confirm the efficiency of the
asynchronous multisplitting algorithm compared to the synchronous GMRES
especially in case of geographically distant clusters.
This paper is organized as follows. Section~\ref{sec:asynchro} presents the
iteration model we use and more particularly the asynchronous scheme. In
-section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
+Section~\ref{sec:simgrid} the SimGrid simulation toolkit is presented.
Section~\ref{sec:04} details the different solvers that we use. Finally our
-experimental results are presented in section~\ref{sec:expe} followed by some
+experimental results are presented in Section~\ref{sec:expe} followed by some
concluding remarks and perspectives.
\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
-Asynchronous iterative methods have been studied for many years theoritecally and
+Asynchronous iterative methods have been studied for many years theoretically and
practically. Many methods have been considered and convergence results have been
proved. These methods can be used to solve, in parallel, fixed point problems
(i.e. problems for which the solution is $x^\star =f(x^\star)$. In practice,
-asynchronous iterations methods can be used to solve, for example, linear and
+asynchronous iteration methods can be used to solve, for example, linear and
non-linear systems of equations or optimization problems, interested readers are
invited to read~\cite{BT89,bahi07}.
requires very few modifications to be able to be executed in both variants. In
practice, only the communications and convergence detection are different. In
the synchronous mode, iterations are synchronized whereas in the asynchronous
-one, they are not. It should be noticed that non blocking communications can be
+one, they are not. It should be noticed that non-blocking communications can be
used in both modes. Concerning the convergence detection, synchronous variants
can use a global convergence procedure which acts as a global synchronization
point. In the asynchronous model, the convergence detection is more tricky as
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
The number of iterations required to reach the convergence is generally greater
-for the asynchronous scheme (this number depends depends on the delay of the
+for the asynchronous scheme (this number depends on the delay of the
messages). Note that, it is not the case in the synchronous mode where the
number of iterations is the same than in the sequential mode. In this way, the
set of the parameters of the platform (number of nodes, power of nodes,
-inter and intra clusters bandwidth and latency, \ldots) and of the
+inter and intra clusters bandwidth and latency,~\ldots) and of the
application can drastically change the number of iterations required to get the
convergence. It follows that asynchronous iterative algorithms are difficult to
optimize since the financial and deployment costs on large scale multi-core
-architecture are often very important. So, prior to delpoyment and tests it
+architectures are often very important. So, prior to deployment and tests it
seems very promising to be able to simulate the behavior of asynchronous
-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced
by simulation are in accordance with reality i.e. of the same order of
magnitude. To our knowledge, there is no study on this problematic.
\hline
Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- - & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
+ Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
+ - & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
\end{tabular}
\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}}
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
- \caption{Various grid configurations with the input matrix size N$_{x}$=150 and N$_{x}$=170\RC{idem}
+ \caption{Various grid configurations with the input matrix size $N_{x}=150$ and $N_{x}=170$\RC{idem}
\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}}
\label{fig:01}
\end{figure}
Grid Architecture & 2x16, 4x8\\ %\hline
Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
- & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
\end{tabular}
\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
\label{tab:02}
\hline
Grid Architecture & 2x16\\ %\hline
Network & N1 : bw=1Gbs \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\caption{Test conditions: network latency impacts}
\label{tab:03}
\hline
Grid Architecture & 2x16\\ %\hline
Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline \\
+ Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
\end{tabular}
\caption{Test conditions: Network bandwidth impacts\RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}}
\label{tab:04}
\hline
Grid Architecture & 4x8\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
+ Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
\end{tabular}
\caption{Test conditions: Input matrix size impacts}
\label{tab:05}
\hline
Grid architecture & 2x16\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
+ Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
\end{tabular}
\caption{Test conditions: CPU Power impacts}
\label{tab:06}
Processors Power & 1 GFlops to 1.5 GFlops\\
Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
- Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
+ Input matrix size & $N_{x}$ = From 62 to 150\\ %\hline
Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
\end{tabular}
\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
converge and so to very different execution times.
-Our future works...
+In future works, we plan to investigate how to simulate the behavior of really
+large scale applications. For example, if we are interested to simulate the
+execution of the solvers of this paper with thousand or even dozens of thousands
+or core, it is not possible to do that with SimGrid. In fact, this tool will
+make the real computation. So we plan to focus our research on that problematic.
