\end{figure}
-The execution time difference between the two algorithms is important when
-comparing between different grid architectures, even with the same number of
-processors (like 2x16 and 4x8 = 32 processors for example). The
-experiment concludes the low sensitivity of the multisplitting method
-(compared with the classical GMRES) when scaling up the number of the processors in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs 40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.
-
-\textit{\\3.b Running on two different speed cluster inter-networks\\}
+The execution times between the two algorithms is significant with different
+grid architectures, even with the same number of processors (for example, 2x16
+and 4x8). We can observ the low sensitivity of the Krylov multisplitting method
+(compared with the classical GMRES) when scaling up the number of the processors
+in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
+40\% better (resp. 48\%) less when running from 2x16=32 to 8x8=64 processors.
+
+\subsubsection{Running on two different speed cluster inter-networks}
+\ \\
-% environment
-\begin{footnotesize}
+\begin{figure} [ht!]
+\begin{center}
\begin{tabular}{r c }
\hline
Grid & 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$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-Table 2 : Clusters x Nodes - Networks N1 x N2 \\
-
- \end{footnotesize}
+\caption{Clusters x Nodes - Networks N1 x N2}
+\end{center}
+\end{figure}
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Cluster x Nodes N1 x N2}
-%\label{overflow}}
+\label{fig:02}
\end{figure}
%\end{wrapfigure}
-The experiments compare the behavior of the algorithms running first on
-a speed inter- cluster network (N1) and also on a less performant network (N2).
-Figure 4 shows that end users will gain to reduce the execution time
-for both algorithms in using a grid architecture like 4x16 or 8x8: the
-performance was increased in a factor of 2. The results depict also that
-when the network speed drops down (12.5\%), the difference between the execution
-times can reach more than 25\%.
+These experiments compare the behavior of the algorithms running first on a
+speed inter-cluster network (N1) and also on a less performant network (N2).
+Figure~\ref{fig:02} shows that end users will gain to reduce the execution time
+for both algorithms in using a grid architecture like 4x16 or 8x8: the
+performance was increased in a factor of 2. The results depict also that when
+the network speed drops down (12.5\%), the difference between the execution
+times can reach more than 25\%. \RC{c'est pas clair : la différence entre quoi et quoi?}
-\textit{\\3.c Network latency impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network latency impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
\begin{tabular}{r c }
\hline
Grid & 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$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ \hline
\end{tabular}
-Table 3 : Network latency impact \\
-
-\end{footnotesize}
+\caption{Network latency impact}
+\end{figure}
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
\caption{Network latency impact on execution time}
-%\label{overflow}}
+\label{fig:03}
\end{figure}
-According the results in figure 5, degradation of the network
-latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time
-increase more than 75\% (resp. 82\%) of the execution for the classical
-GMRES (resp. multisplitting) algorithm. In addition, it appears that the
-multisplitting method tolerates more the network latency variation with
-a less rate increase of the execution time. Consequently, in the worst case (lat=6.10$^{-5
-}$), the execution time for GMRES is almost the double of the time for
-the multisplitting, even though, the performance was on the same order
-of magnitude with a latency of 8.10$^{-6}$.
+According the results in Figure~\ref{fig:03}, a degradation of the network
+latency from 8.10$^{-6}$ to 6.10$^{-5}$ implies an absolute time increase more
+than 75\% (resp. 82\%) of the execution for the classical GMRES (resp. Krylov
+multisplitting) algorithm. In addition, it appears that the Krylov
+multisplitting method tolerates more the network latency variation with a less
+rate increase of the execution time. Consequently, in the worst case
+(lat=6.10$^{-5 }$), the execution time for GMRES is almost the double than the
+time of the Krylov multisplitting, even though, the performance was on the same
+order of magnitude with a latency of 8.10$^{-6}$.
-\textit{\\3.d Network bandwidth impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Network bandwidth impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
\begin{tabular}{r c }
\hline
Grid & 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 \\
\end{tabular}
-Table 4 : Network bandwidth impact \\
-
-\end{footnotesize}
+\caption{Network bandwidth impact}
+\end{figure}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
\caption{Network bandwith impact on execution time}
-%\label{overflow}
+\label{fig:04}
\end{figure}
-The results of increasing the network bandwidth show the improvement
-of the performance for both of the two algorithms by reducing the execution time (Figure 6). However, and again in this case, the multisplitting method presents a better performance in the considered bandwidth interval with a gain of 40\% which is only around 24\% for classical GMRES.
+The results of increasing the network bandwidth show the improvement of the
+performance for both algorithms by reducing the execution time (see
+Figure~\ref{fig:04}). However, in this case, the Krylov multisplitting method
+presents a better performance in the considered bandwidth interval with a gain
+of 40\% which is only around 24\% for classical GMRES.
-\textit{\\3.e Input matrix size impacts on performance\\}
-
-% environment
-\begin{footnotesize}
+\subsubsection{Input matrix size impacts on performance}
+\ \\
+\begin{figure} [ht!]
+\centering
\begin{tabular}{r c }
\hline
Grid & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ = From 40 to 200\\ \hline \\
+ Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
+ Input matrix size & N$_{x}$ = From 40 to 200\\ \hline
\end{tabular}
-Table 5 : Input matrix size impact\\
-
-\end{footnotesize}
+\caption{Input matrix size impact}
+\end{figure}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Pb size impact on execution time}
-%\label{overflow}}
+\caption{Problem size impact on execution time}
+\label{fig:05}
\end{figure}
-In this experimentation, the input matrix size has been set from
-N$_{x}$ = N$_{y}$ = N$_{z}$ = 40 to 200 side elements that is from 40$^{3}$ = 64.000 to
-200$^{3}$ = 8.000.000 points. Obviously, as shown in the figure 7,
-the execution time for the two algorithms convergence increases with the
-iinput matrix size. But the interesting results here direct on (i) the
-drastic increase (300 times) of the number of iterations needed before
-the convergence for the classical GMRES algorithm when the matrix size
-go beyond N$_{x}$=150; (ii) the classical GMRES execution time also almost
-the double from N$_{x}$=140 compared with the convergence time of the
-multisplitting method. These findings may help a lot end users to setup
-the best and the optimal targeted environment for the application
-deployment when focusing on the problem size scale up. Note that the
-same test has been done with the grid 2x16 getting the same conclusion.
-
-\textit{\\3.f CPU Power impact on performance\\}
+In these experiments, the input matrix size has been set from N$_{x}$ = N$_{y}$
+= N$_{z}$ = 40 to 200 side elements that is from 40$^{3}$ = 64.000 to 200$^{3}$
+= 8,000,000 points. Obviously, as shown in Figure~\ref{fig:05}, the execution
+time for both algorithms increases when the input matrix size also increases.
+But the interesting results are:
+\begin{enumerate}
+ \item the drastic increase (300 times) \RC{Je ne vois pas cela sur la figure}
+of the number of iterations needed to reach the convergence for the classical
+GMRES algorithm when the matrix size go beyond N$_{x}$=150;
+\item the classical GMRES execution time is almost the double for N$_{x}$=140
+ compared with the Krylov multisplitting method.
+\end{enumerate}
-% environment
-\begin{footnotesize}
+These findings may help a lot end users to setup the best and the optimal
+targeted environment for the application deployment when focusing on the problem
+size scale up. It should be noticed that the same test has been done with the
+grid 2x16 leading to the same conclusion.
+
+\subsubsection{CPU Power impact on performance}
+
+\begin{figure} [ht!]
+\centering
\begin{tabular}{r c }
\hline
Grid & 2x16\\ %\hline
Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
Input matrix size & N$_{x}$ = 150 x 150 x 150\\ \hline
\end{tabular}
-Table 6 : CPU Power impact \\
-
-\end{footnotesize}
-
+\caption{CPU Power impact}
+\end{figure}
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
\caption{CPU Power impact on execution time}
-%\label{overflow}}
-s\end{figure}
-
-Using the Simgrid simulator flexibility, we have tried to determine the
-impact on the algorithms performance in varying the CPU power of the
-clusters nodes from 1 to 19 GFlops. The outputs depicted in the figure 6
-confirm the performance gain, around 95\% for both of the two methods,
-after adding more powerful CPU.
-
-\subsection{Comparing GMRES in native synchronous mode and
-Multisplitting algorithms in asynchronous mode}
-
-The previous paragraphs put in evidence the interests to simulate the
-behavior of the application before any deployment in a real environment.
-We have focused the study on analyzing the performance in varying the
-key factors impacting the results. The study compares
-the performance of the two proposed algorithms both in \textit{synchronous mode
-}. In this section, following the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textit{
-asynchronous mode} compared with the classical GMRES staying in
-\textit{synchronous mode}.
-
-Note that the interest of using the asynchronous mode for data exchange
-is mainly, in opposite of the synchronous mode, the non-wait aspects of
-the current computation after a communication operation like sending
-some data between nodes. Each processor can continue their local
-calculation without waiting for the end of the communication. Thus, the
-asynchronous may theoretically reduce the overall execution time and can
-improve the algorithm performance.
-
-As stated supra, Simgrid simulator tool has been used to prove the
-efficiency of the multisplitting in asynchronous mode and to find the
-best combination of the grid resources (CPU, Network, input matrix size,
-\ldots ) to get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ / exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+\label{fig:06}
+\end{figure}
+
+Using the Simgrid simulator flexibility, we have tried to determine the impact
+on the algorithms performance in varying the CPU power of the clusters nodes
+from 1 to 19 GFlops. The outputs depicted in Figure~\ref{fig:06} confirm the
+performance gain, around 95\% for both of the two methods, after adding more
+powerful CPU.
+
+\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+
+The previous paragraphs put in evidence the interests to simulate the behavior
+of the application before any deployment in a real environment. In this
+section, following the same previous methodology, our goal is to compare the
+efficiency of the multisplitting method in \textit{ asynchronous mode} with the
+classical GMRES in \textit{synchronous mode}.
+
+The interest of using an asynchronous algorithm is that there is no more
+synchronization. With geographically distant clusters, this may be essential.
+In this case, each processor can compute its iteration freely without any
+synchronization with the other processors. Thus, the asynchronous may
+theoretically reduce the overall execution time and can improve the algorithm
+performance.
+
+\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
+As stated before, the Simgrid simulator tool has been successfully used to show
+the efficiency of the multisplitting in asynchronous mode and to find the best
+combination of the grid resources (CPU, Network, input matrix size, \ldots ) to
+get the highest \textit{"relative gain"} (exec\_time$_{GMRES}$ /
+exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
The test conditions are summarized in the table below : \\
-% environment
-\begin{footnotesize}
+\begin{figure} [ht!]
+\centering
\begin{tabular}{r c }
\hline
Grid & 2x50 totaling 100 processors\\ %\hline
Input matrix size & N$_{x}$ = From 62 to 150\\ %\hline
Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
\end{tabular}
-\end{footnotesize}
+\end{figure}
-Again, comprehensive and extensive tests have been conducted varying the
-CPU power and the network parameters (bandwidth and latency) in the
-simulator tool with different problem size. The relative gains greater
-than 1 between the two algorithms have been captured after each step of
-the test. Table 7 below has recorded the best grid configurations
-allowing the multisplitting method execution time more performant 2.5 times than
-the classical GMRES execution and convergence time. The experimentation has demonstrated the relative multisplitting algorithm tolerance when using a low speed network that we encounter usually with distant clusters thru the internet.
+Again, comprehensive and extensive tests have been conducted with different
+parametes as the CPU power, the network parameters (bandwidth and latency) in
+the simulator tool and with different problem size. The relative gains greater
+than 1 between the two algorithms have been captured after each step of the
+test. In Figure~\ref{table:01} are reported the best grid configurations
+allowing the multisplitting method to be more than 2.5 times faster than the
+classical GMRES. These experiments also show the relative tolerance of the
+multisplitting algorithm when using a low speed network as usually observed with
+geographically distant clusters throuth the internet.
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\end{tabular}}
-\begin{table}[!t]
- \centering
+\begin{figure}[!t]
+\centering
+%\begin{table}
% \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
% \label{"Table 7"}
-Table 7. Relative gain of the multisplitting algorithm compared with
-the classical GMRES \\
-
- \begin{mytable}{11}
+ \begin{mytable}{11}
\hline
bandwidth (Mbit/s)
& 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
& 2.52 & 2.55 & 2.52 & 2.57 & 2.54 & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\
\hline
\end{mytable}
-\end{table}
+%\end{table}
+ \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
+ \label{table:01}
+\end{figure}
+
\section{Conclusion}
CONCLUSION
