\subsection{Comparison between GMRES and two-stage multisplitting algorithms in synchronous mode}
In the scope of this paper, our first objective is to analyze when the synchronous Krylov two-stage method has better performance than the classical GMRES method. With a synchronous iterative method, better performance means a smaller number of iterations and execution time before reaching the convergence.
-Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$. In what follows, we will present the test conditions, the output results and our comments.
+Table~\ref{tab:01} summarizes the parameters used in the different simulations: the grid architectures, the network of inter-clusters backbone links and the matrix sizes of the 3D Poisson problem. However, for all simulations we fix the network parameters of the intra-clusters links: the bandwidth $bw$=10Gbs and the latency $lat=8\mu$s. In what follows, we will present the test conditions, the output results and our comments.
\begin{table} [ht!]
\begin{center}
\begin{tabular}{ll}
\hline
Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
- & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+ & $N2$: $bw$=1Gbs, $lat=50\mu$s \\
\multirow{2}{*}{Matrix size} & $Mat1$: N$_{x}\times$N$_{y}\times$N$_{z}$=150$\times$150$\times$150\\
& $Mat2$: N$_{x}\times$N$_{y}\times$N$_{z}$=170$\times$170$\times$170 \\ \hline
\end{tabular}
The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16).
-\begin{figure}[t]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
-\LZK{CE, la légende de la Figure 3 est trop large. Remplacer les N$_x\times$N$_y\times$N$_z$ par $Mat1$=150$^3$ et $Mat2$=170$^3$ comme dans la Table 1}
\label{fig:01}
\end{figure}
\subsubsection{Simulations for two different inter-clusters network speeds\\}
-In Figure~\ref{fig:02} we present the execution times of both algorithms to solve a 3D Poisson problem of size $150^3$ on two different simulated network $N1$ and $N2$ (see Table~\ref{tab:01}). As it was previously said, we can see from the figure that the Krylov two-stage algorithm is more sensitive the number of clusters than the GMRES algorithm. However, we can notice an interesting behavior of the Krylov two-stage algorithm. It is less sensitive to bad network bandwidth and latency for the inter-clusters links than the GMRES algorithms. This means that the multisplitting methods are more efficient for distributed systems with high latency networks.
-
-
-
-
-%% The figure shows that the Krylov two-stage algorithm is more sensitive the number of clusters than the GMRES algorithm.
-
-%% In this section, the experiments compare the behavior of the algorithms running on a
-%% speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}.
-%% %\RC{Il faut définir cela avant...}
-%% Figure~\ref{fig:02} shows that end users will reduce the execution time
-%% for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: the reduction factor is around $2$. The results depict also that when
-%% the network speed drops down (variation of 12.5\%), the difference between the two Multisplitting algorithms execution times can reach more than 25\%.
-
-\LZK{J'ai mis que le problème résolu dans la figure 4 est de taille $150^3$. CE, pourrais tu le confirmer?}
-
-\begin{figure}[t]
+In Figure~\ref{fig:02} we present the execution times of both algorithms to
+solve a 3D Poisson problem of size $150^3$ on two different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure that the Krylov two-stage algorithm is sensitive to the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an interesting behavior of the Krylov two-stage algorithm. It is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the GMRES algorithms. This means that the multisplitting methods are more
+efficient for distributed systems with high latency networks.
+
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
\caption{Various grid configurations with networks $N1$ vs. $N2$}
\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+\RCE{ok}
\label{fig:02}
\end{figure}
+\subsubsection{Network latency impacts on performances\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm.
-
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-
-\subsubsection{Network latency impacts on performance\\}
-
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2 $\times$ 16\\ %\hline
- \multirow{2}{*}{Inter Network N1} & $bw$=1Gbs, \\ %\hline
- & $lat$= From 8$\times$10$^{-6}$ to $6.10^{-5}$ second \\
- 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}
-\end{table}
-
-\begin{figure} [htbp]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impacts on execution time}
-%\AG{\np{E-6}}}
+\caption{Network latency impacts on performances}
\label{fig:03}
\end{figure}
-In Table~\ref{tab:03}, parameters for the influence of the network latency are
-reported. According to the results of Figure~\ref{fig:03}, a degradation of the
-network latency from $8.10^{-6}$ to $6.10^{-5}$ implies an absolute time
-increase of more than $75\%$ (resp. $82\%$) of the execution for the classical
-GMRES (resp. Krylov multisplitting) algorithm. The execution time factor
-between the two algorithms varies from 2.2 to 1.5 times with a network latency
-decreasing from $8.10^{-6}$ to $6.10^{-5}$ second.
-
+\subsubsection{Network bandwidth impacts on performances\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
-\subsubsection{Network bandwidth impacts on performance\\}
-
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2 $\times$ 16\\ %\hline
-\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline
- & $lat$= 5.10$^{-5}$ second \\
- 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}
-%\RCE{C est le bw}
-\label{tab:04}
-\end{table}
-
-
-\begin{figure} [htbp]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impacts on execution time}
-%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
-%\RCE{Corrige}
+\caption{Network bandwith impacts on performances}
\label{fig:04}
\end{figure}
-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 the classical GMRES.
-
-\subsubsection{Input matrix size impacts on performance\\}
-
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 4 $\times$ 8\\ %\hline
- Inter Network & $bw$=1Gbs - $lat$=5.10$^{-5}$ \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline
- \end{tabular}
-\caption{Test conditions: Input matrix size impacts}
-\label{tab:05}
-\end{table}
-
+\subsubsection{Matrix size impacts on performances\\}
+In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes. For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
+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.
-\begin{figure} [htbp]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Problem size impacts on execution time}
+\caption{Problem size impacts on performances}
\label{fig:05}
\end{figure}
-In these experiments, the input matrix size has been set from $50^3$ to
-$190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both
-algorithms increases when the input matrix size also increases. For all problem
-sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this
-benchmark, it seems that the greater the problem size is, the bigger the ratio
-between both algorithm execution times is. We can also observ that for some
-problem sizes, the Krylov multisplitting convergence varies quite a
-lot. Consequently the execution times in that cases also varies.
-
-
-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 4 $\times$ 8 leading to the same conclusion.
-
-\subsubsection{CPU Power impacts on performance\\}
-
-
-\begin{table} [htbp]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid architecture & 2 $\times$ 16\\ %\hline
- Inter Network & N2 : $bw$=1Gbs - $lat$=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} = 150 \times 150 \times 150$\\
- CPU Power & From 3 to 19 GFlops \\ \hline
- \end{tabular}
-\caption{Test conditions: CPU Power impacts}
-\label{tab:06}
-\end{table}
+\subsubsection{CPU power impacts on performances\\}
+Using the SimGrid simulator flexibility, we have tried to determine the impact of the CPU power of the processors in the different clusters on performances of both algorithms. We have varied the CPU power from $1$GFlops to $19$GFlops. The simulation is conducted in a grid of 2$\times$16 processors interconnected by the network $N2$ (see Table~\ref{tab:01}) to solve a 3D Poisson problem of size $150^3$. The results depicted in Figure~\ref{fig:06} confirm the performance gain, about $95\%$ for both algorithms, after improving the CPU power of processors.
-\begin{figure} [ht!]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cpu_power_impact_on_execution_time.pdf}
-\caption{CPU Power impacts on execution time}
+\caption{CPU Power impacts on performances}
\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.
\ \\
-%\DL{il faut une conclusion sur ces tests : ils confirment les résultats déjà
-%obtenus en grandeur réelle. Donc c'est une aide précieuse pour les dev. Pas
-%besoin de déployer sur une archi réelle}
-
To conclude these series of experiments, with SimGrid we have been able to make
many simulations with many parameters variations. Doing all these experiments
with a real platform is most of the time not possible. Moreover the behavior of
-both GMRES and Krylov multisplitting methods is in accordance with larger real
-executions on large scale supercomputer~\cite{couturier15}.
+both GMRES and Krylov two-stage algorithms is in accordance with larger real
+executions on large scale supercomputers~\cite{couturier15}.
-\subsection{Comparing GMRES in native synchronous mode and the multisplitting algorithm in asynchronous mode}
+\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
The previous paragraphs put in evidence the interests to simulate the behavior
of the application before any deployment in a real environment. In this
theoretically reduce the overall execution time and can improve the algorithm
performance.
-In this section, the Simgrid simulator is used to compare the behavior of the
-multisplitting in asynchronous mode with GMRES in synchronous mode. Several
-benchmarks have been performed with various combination of the grid resources
-(CPU, Network, input matrix size, \ldots ). The test conditions are summarized
-in Table~\ref{tab:07}. In order to compare the execution times, this table
+In this section, the SimGrid simulator is used to compare the behavior of the
+two-stage algorithm in asynchronous mode with GMRES in synchronous mode. Several
+benchmarks have been performed with various combinations of the grid resources
+(CPU, Network, matrix size, \ldots). The test conditions are summarized
+in Table~\ref{tab:02}. In order to compare the execution times, Table~\ref{tab:03}
reports the relative gain between both algorithms. It is defined by the ratio
between the execution time of GMRES and the execution time of the
-multisplitting. The ration is greater than one because the asynchronous
+multisplitting.
+\LZK{Quelle table repporte les gains relatifs?? Sûrement pas Table II !!}
+\RCE{Table III avec la nouvelle numerotation}
+The ratio is greater than one because the asynchronous
multisplitting version is faster than GMRES.
-
-
-\begin{table} [htbp]
+\begin{table}[htbp]
\centering
-\begin{tabular}{r c }
+\begin{tabular}{ll}
\hline
- Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline
- 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
- Residual error precision & 10$^{-5}$ to 10$^{-9}$\\ \hline \\
+ Grid architecture & 2$\times$50 totaling 100 processors\\
+ Processors Power & 1 GFlops to 1.5 GFlops \\
+ \multirow{2}{*}{Network inter-clusters} & $bw$=1.25 Gbits, $lat=50\mu$s \\
+ & $bw$=5 Mbits, $lat=20ms$s\\
+ Matrix size & from $62^3$ to $150^3$\\
+ Residual error precision & $10^{-5}$ to $10^{-9}$\\ \hline \\
\end{tabular}
-\caption{Test conditions: GMRES in synchronous mode vs Krylov Multisplitting in asynchronous mode}
-\label{tab:07}
+\caption{Test conditions: GMRES in synchronous mode vs. Krylov two-stage in asynchronous mode}
+\label{tab:02}
\end{table}
-Again, comprehensive and extensive tests have been conducted with different
-parameters as the CPU power, the network parameters (bandwidth and latency)
-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
-Table~\ref{tab:08} 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 through the internet.
% use the same column width for the following three tables
\newlength{\mytablew}\settowidth{\mytablew}{\footnotesize\np{E-11}}
\hline
\end{mytable}
%\end{table}
- \caption{Relative gain of the multisplitting algorithm compared with the classical GMRES}
- \label{tab:08}
+ \caption{Relative gains of the two-stage multisplitting algorithm compared with the classical GMRES}
+ \label{tab:03}
\end{table}
+Again, comprehensive and extensive tests have been conducted with different
+parameters as the CPU power, the network parameters (bandwidth and latency)
+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
+Table~\ref{tab:08} are reported the best grid configurations allowing
+the two-stage multisplitting algorithm 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 through the internet.
+
\section{Conclusion}
-
In this paper we have presented the simulation of the execution of three
-different parallel solvers on some multi-core architectures. We have show that
+different parallel solvers on some multi-core architectures. We have shown that
the SimGrid toolkit is an interesting simulation tool that has allowed us to
determine which method to choose given a specified multi-core architecture.
Moreover the simulated results are in accordance (i.e. with the same order of
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
+of cores, 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.
