\end{array} \right)
\end{equation*}
in such a way that successive rows of matrix $A$ and both vectors $x$ and $b$
-are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$ $A_{\ell
+are assigned to one cluster, where for all $\ell,m\in\{1,\ldots,L\}$, $A_{\ell
m}$ is a rectangular block of $A$ of size $n_\ell\times n_m$, $X_\ell$ and
-$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each and
-$\sum_{\ell} n_\ell=\sum_{m} n_m=n$.
+$B_\ell$ are sub-vectors of $x$ and $b$, respectively, of size $n_\ell$ each,
+and $\sum_{\ell} n_\ell=\sum_{m} n_m=n$.
The multisplitting method proceeds by iteration to solve in parallel the linear system on $L$ clusters of processors, in such a way each sub-system
\begin{equation}
study that the results depend on the following parameters:
\begin{itemize}
\item At the network level, we found that the most critical values are the
- bandwidth (bw) and the network latency (lat).
+ bandwidth and the network latency.
\item Hosts power (GFlops) can also influence on the results.
\item Finally, when submitting job batches for execution, the arguments values
- passed to the program like the maximum number of iterations or the
- \textit{external} precision are critical. They allow to ensure not only the
- convergence of the algorithm but also to get the main objective of the
- experimentation of the simulation in having an execution time in asynchronous
- less than in synchronous mode. The ratio between the execution time of asynchronous compared to the synchronous mode is defined as the "relative gain". So, our objective running the algorithm in SimGrid is to obtain a relative gain greater than 1.
+ passed to the program like the maximum number of iterations or the external
+ precision are critical. They allow to ensure not only the convergence of the
+ algorithm but also to get the main objective of the experimentation of the
+ simulation in having an execution time in asynchronous less than in
+ synchronous mode. The ratio between the execution time of asynchronous
+ compared to the synchronous mode is defined as the \emph{relative gain}. So,
+ our objective running the algorithm in SimGrid is to obtain a relative gain
+ greater than 1.
+ \AG{$t_\text{async} / t_\text{sync} > 1$, l'objectif est donc que ça dure plus
+ longtemps (que ça aille moins vite) en asynchrone qu'en synchrone ?
+ Ce n'est pas plutôt l'inverse ?}
\end{itemize}
-A priori, obtaining a relative gain greater than 1 would be difficult in a local area
-network configuration where the synchronous mode will take advantage on the
+A priori, obtaining a relative gain greater than 1 would be difficult in a local
+area network configuration where the synchronous mode will take advantage on the
rapid exchange of information on such high-speed links. Thus, the methodology
adopted was to launch the application on clustered network. In this last
-configuration, degrading the inter-cluster network performance will
-\textit{penalize} the synchronous mode allowing to get a relative gain greater than 1.
-This action simulates the case of distant clusters linked with long distance network
-like Internet.
-
+configuration, degrading the inter-cluster network performance will penalize the
+synchronous mode allowing to get a relative gain greater than 1. This action
+simulates the case of distant clusters linked with long distance network like
+Internet.
+
+\AG{Cette partie sur le poisson 3D
+ % on sait donc que ce n'est pas une plie ou une sole (/me fatigué)
+ n'est pas à sa place. Elle devrait être placée plus tôt.}
In this paper, we solve the 3D Poisson problem whose the mathematical model is
\begin{equation}
\left\{
As a first step, the algorithm was run on a network consisting of two clusters
containing 50 hosts each, totaling 100 hosts. Various combinations of the above
-factors have providing the results shown in Table~\ref{tab.cluster.2x50} with a
+factors have provided the results shown in Table~\ref{tab.cluster.2x50} with a
matrix size ranging from $N_x = N_y = N_z = \text{62}$ to 171 elements or from
$\text{62}^\text{3} = \text{\np{238328}}$ to $\text{171}^\text{3} =
\text{\np{5000211}}$ entries.
\begin{mytable}{6}
\hline
- bw
+ bandwidth
& 5 & 5 & 5 & 5 & 5 & 50 \\
\hline
- lat
+ latency
& 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
\hline
power
& 62 & 62 & 62 & 100 & 100 & 110 \\
\hline
Prec/Eprec
- & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} & \np{E-11} \\
+ \hline
\hline
Relative gain
& 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 \\
\hline
\end{mytable}
- \smallskip
+ \bigskip
\begin{mytable}{6}
\hline
- bw
+ bandwidth
& 50 & 50 & 50 & 50 & 10 & 10 \\
\hline
- lat
+ latency
& 0.02 & 0.02 & 0.02 & 0.02 & 0.03 & 0.01 \\
\hline
power
Prec/Eprec
& \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-5} & \np{E-5} \\
\hline
+ \hline
Relative gain
& 2.51 & 2.58 & 2.55 & 2.54 & 1.59 & 1.29 \\
\hline
\begin{mytable}{6}
\hline
- bw
+ bandwidth
& 10 & 5 & 4 & 3 & 2 & 6 \\
\hline
- lat
+ latency
& 0.01 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\
\hline
power
Prec/Eprec
& \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} & \np{E-5} \\
\hline
+ \hline
Relative gain
& 1.003 & 1.01 & 1.08 & 1.19 & 1.28 & 1.01 \\
\hline
\begin{mytable}{1}
\hline
- bw & 1 \\
+ bandwidth & 1 \\
\hline
- lat & 0.02 \\
+ latency & 0.02 \\
\hline
power & 1 \\
\hline
\hline
Prec/Eprec & \np{E-5} \\
\hline
+ \hline
Relative gain & 1.11 \\
\hline
\end{mytable}
~\\{}\AG{Donner un peu plus de précisions (plateforme en particulier).}
\begin{itemize}
- \item HOSTFILE: Hosts file description.
- \item PLATFORM: file description of the platform architecture : clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-lat latency, \dots{}).
+\item HOSTFILE: Hosts file description.
+\item PLATFORM: file description of the platform architecture : clusters (CPU
+ power, \dots{}), intra cluster network description, inter cluster network
+ (bandwidth, latency, \dots{}).
\end{itemize}
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[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5 is obtained with
+\np[Mbit/s]{50}, the result of efficiency with a relative gain of 1.5\AG[]{2.5 ?} is obtained with
high external precision of \np{E-11} for a matrix size from 110 to 150 side
elements.
(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 ?}
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
\RC{Est ce qu'on sait expliquer pourquoi il y a une telle différence entre les résultats avec 2 et 3 clusters... Avec 3 clusters, ils sont pas très bons... Je me demande s'il ne faut pas les enlever...}
\RC{En fait je pense avoir la réponse à ma remarque... On voit avec les 2 clusters que le gain est d'autant plus grand qu'on choisit une bonne précision. Donc, plusieurs solutions, lancer rapidement un long test pour confirmer ca, ou enlever des tests... ou on ne change rien :-)}
-\LZK{Ma question est: le bw et lat sont ceux inter-clusters ou pour les deux inter et intra cluster??}
+\LZK{Ma question est: le bandwidth et latency sont ceux inter-clusters ou pour les deux inter et intra cluster??}
\section{Conclusion}
The experimental results on executing a parallel iterative algorithm in
% 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
+% LocalWords: intra durations nonsingular Waitall discretization discretized
+% LocalWords: InnerSolver Isend Irecv
