\subsection{Simulation of two-stage methods using SimGrid framework}
\label{sec:04.02}
-One of our objectives when simulating the application in SIMGRID is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in SIMGRID simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.
+One of our objectives when simulating the application in Simgrid is, as in real life, to get accurate results (solutions of the problem) but also ensure the test reproducibility under the same conditions. According our experience, very few modifications are required to adapt a MPI program to run in Simgrid simulator using SMPI (Simulator MPI).The first modification is to include SMPI libraries and related header files (smpi.h). The second and important modification is to eliminate all global variables in moving them to local subroutine or using a Simgrid selector called "runtime automatic switching" (smpi/privatize\_global\_variables). Indeed, global variables can generate side effects on runtime between the threads running in the same process, generated by the Simgrid to simulate the grid environment.The last modification on the MPI program pointed out for some cases, the review of the sequence of the MPI\_Isend, MPI\_Irecv and MPI\_Waitall instructions which might cause an infinite loop.
-\paragraph{SIMGRID Simulator parameters}
+\paragraph{Simgrid Simulator parameters}
\begin{itemize}
\item hostfile: Hosts description file.
\textbf{Step 2} : Collect the software materials needed for the
experimentation. In our case, we have two variants algorithms for the
-resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, SIMGRID simulator has been chosen to simulate the behaviors of the
-distributed applications. SIMGRID is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
+resolution of three 3D-Poisson problem: (1) using the classical GMRES (Algo-1)(2) and the multisplitting method (Algo-2). In addition, Simgrid simulator has been chosen to simulate the behaviors of the
+distributed applications. Simgrid is running on the Mesocentre datacenter in Franche-Comte University but also in a virtual machine on a laptop. \\
\textbf{Step 3} : Fix the criteria which will be used for the future
results comparison and analysis. In the scope of this study, we retain
In the scope of this paper, our first objective is to demonstrate the
Algo-2 (Multisplitting method) shows a better performance in grid
architecture compared with Algo-1 (Classical GMRES) both running in
-\textbf{\textit{synchronous mode}}. Better algorithm performance
+\textit{synchronous mode}. Better algorithm performance
should means a less number of iterations output and a less execution time
before reaching the convergence. For a systematic study, the experiments
should figure out that, for various grid parameters values, the
\begin{tabular}{r c }
\hline
Grid & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5E-05 \\ %\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 \\
%\label{overflow}}
\end{figure}
-Using the SIMGRID simulator flexibility, we have tried to determine the
+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. Note that the execution time axis in the
-figure is in logarithmic scale.
+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. In the same line, the study compares
-the performance of the two proposed methods in \textbf{synchronous mode
-}. In this section, with the same previous methodology, the goal is to
-demonstrate the efficiency of the multisplitting method in \textbf{
-asynchronous mode} compare with the classical GMRES staying in the
-synchronous mode.
+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
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
+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 "\,relative gain" in comparison with the
-classical GMRES time.
+\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 : \\
\hline
Grid & 2x50 totaling 100 processors\\ %\hline
Processors & 1 GFlops to 1.5 GFlops\\
- Intra-Network & bw=1.25 Gbits - lat=5E-05 \\ %\hline
- Inter-Network & bw=5 Mbits - lat=2E-02\\
+ 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
- Residual error precision: 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+ Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
\end{tabular}
\end{footnotesize}
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 I below has recorded the best grid configurations
-allowing a multiplitting method time more than 2.5 times lower than
-classical GMRES execution and convergence time. The finding thru this
-experimentation is the tolerance of the multisplitting method under a
-low speed network that we encounter usually with distant clusters thru the
-internet.
+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.
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
|*{#1}{>{\centering\arraybackslash}p{\mytablew}|}}}{%
\end{tabular}}
+
\begin{table}[!t]
\centering
- \caption{Relative gain of the multisplitting algorithm compared with
-the classical GMRES}
- \label{"Table 7"}
-
- \begin{mytable}{6}
- \hline
- bandwidth (Mbit/s)
- & 5 & 5 & 5 & 5 & 5 \\
- \hline
- latency (ms)
- & 20 & 20 & 20 & 20 & 20 \\
- \hline
- power (GFlops)
- & 1 & 1 & 1 & 1.5 & 1.5 \\
- \hline
- size (N)
- & 62 & 62 & 62 & 100 & 100 \\
- \hline
- Precision
- & \np{E-5} & \np{E-8} & \np{E-9} & \np{E-11} & \np{E-11} \\
- \hline
- Relative gain
- & 2.52 & 2.55 & 2.52 & 2.57 & 2.54 \\
- \hline
- \end{mytable}
-
- \smallskip
+% \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}{6}
+ \begin{mytable}{11}
\hline
bandwidth (Mbit/s)
- & 50 & 50 & 50 & 50 & 50 \\
+ & 5 & 5 & 5 & 5 & 5 & 50 & 50 & 50 & 50 & 50 \\
\hline
latency (ms)
- & 20 & 20 & 20 & 20 & 20 \\
+ & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 20 \\
\hline
power (GFlops)
- & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
+ & 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
\hline
size (N)
- & 110 & 120 & 130 & 140 & 150 \\
+ & 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
\hline
Precision
- & \np{E-11} & \np{E-11} & \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} & \np{E-11} & \np{E-11} & \np{E-11} & \np{E-11}\\
\hline
Relative gain
- & 2.53 & 2.51 & 2.58 & 2.55 & 2.54 \\
+ & 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}
