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.}
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 \\
\hline
- size $(n^3)$
+ size $(N)$
& 62 & 62 & 62 & 100 & 100 \\
\hline
Precision
Power (GFlops)
& 1.5 & 1.5 & 1.5 & 1.5 & 1.5 \\ % & 1 & 1.5 \\
\hline
- size $(n^3)$
+ size $(N)$
& 110 & 120 & 130 & 140 & 150 \\ % & 171 & 171 \\
\hline
Precision
\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}
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
+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
\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.
