%% help developers to better tune their applications for a given multi-core
%% architecture.
-%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations.
+%% In this paper we focus our attention on the simulation of iterative algorithms to solve sparse linear systems on large clusters. We study the behavior of the widely used GMRES algorithm and two different variants of the Multisplitting algorithms: one using synchronous iterations and another one with asynchronous iterations.
%% For each algorithm we have simulated
%% different architecture parameters to evaluate their influence on the overall
-%% execution time.
+%% execution time.
%% The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous Multisplitting algorithm on distant clusters compared to the synchronous GMRES algorithm.
-The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
+The behavior of multi-core applications is always a challenge to predict, especially with a new architecture for which no experiment has been performed. With some applications, it is difficult, if not impossible, to build accurate performance models. That is why another solution is to use a simulation tool which allows us to change many parameters of the architecture (network bandwidth, latency, number of processors) and to simulate the execution of such applications.
In this paper we focus on the simulation of iterative algorithms to solve sparse linear systems. We study the behavior of the GMRES algorithm and two different variants of the multisplitting algorithms: using synchronous or asynchronous iterations. For each algorithm we have simulated different architecture parameters to evaluate their influence on the overall execution time. The simulations confirm the real results previously obtained on different real multi-core architectures and also confirm the efficiency of the asynchronous multisplitting algorithm on distant clusters compared to the GMRES algorithm.
\end{equation}
until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
-In the parallel context, the 3D Poisson problem is partitioned into $L\times p$ sub-problems such that $L$ is the number of clusters and $p$ is the number of processors in each cluster. We apply the three-dimensional partitioning instead of the row-by-row one in order to reduce the size of the data shared at the sub-problems boundaries. In this case, each processor is in charge of parallelepipedic block of the problem and has at most six neighbors in the same cluster or in distant clusters with which it shares data at boundaries.
+In the parallel context, the 3D Poisson problem is partitioned into $L\times p$
+sub-problems such that $L$ is the number of clusters and $p$ is the number of
+processors in each cluster. We apply the three-dimensional partitioning instead
+of the row-by-row one in order to reduce the size of the data shared at the
+sub-problems boundaries. In this case, each processor is in charge of
+parallelepipedic block of the problem and has at most six neighbors in the same
+cluster or in distant clusters with which it shares data at boundaries.
\subsection{Study setup and simulation methodology}
Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
and between distant clusters. This parameter is application dependent.
- In a grid environment, it is common to distinguish, on the one hand, the
- "intra-network" which refers to the links between nodes within a cluster and
- on the other hand, the "inter-network" which is the backbone link between
- clusters. In practice, these two networks have different speeds.
- The intra-network generally works like a high speed local network with a
- high bandwidth and very low latency. In opposite, the inter-network connects
- clusters sometime via heterogeneous networks components through internet with
- a lower speed. The network between distant clusters might be a bottleneck
- for the global performance of the application.
-
-
-\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.
+ In a grid environment, it is common to distinguish, on one hand, the
+ \textit{intra-network} which refers to the links between nodes within a
+ cluster and on the other hand, the \textit{inter-network} which is the
+ backbone link between clusters. In practice, these two networks have
+ different speeds. The intra-network generally works like a high speed
+ local network with a high bandwidth and very low latency. In opposite, the
+ inter-network connects clusters sometime via heterogeneous networks components
+ through internet with a lower speed. The network between distant clusters
+ might be a bottleneck for the global performance of the application.
+
+
+\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 (i.e. the number of clusters and the number of nodes per
+cluster), 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}$ \\
+Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$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}
In this section, we analyze the simulations conducted on various grid
configurations and for different sizes of the 3D Poisson problem. The parameters
of the network between clusters is fixed to $N2$ (see
-Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and a
-given matrix size 170$^3$ elements, a non-variation in the number of iterations
-for the classical GMRES algorithm, which is not the case of the Krylov two-stage
-algorithm. In fact, with multisplitting algorithms, the number of splitting (in
-our case, it is the number of clusters) influences on the convergence speed. The
-higher the number of splitting is, the slower the convergence of the algorithm
-is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).
-
-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$16).
-
-\begin{figure}[t]
+Table~\ref{tab:01}). Figure~\ref{fig:01} shows, for all grid configurations and
+a given matrix size of 170$^3$ elements, a non-variation in the number of
+iterations for the classical GMRES algorithm, which is not the case of the
+Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number
+of splitting (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splitting is, the slower the
+convergence of the algorithm is (see the output results obtained from
+configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
+8$\times$8).
+
+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$16).
+
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
-\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\caption{Various grid configurations with two matrix sizes: $150^3$ and $170^3$}
\label{fig:01}
\end{figure}
the GMRES algorithms. This means that the multisplitting methods are more
efficient for distributed systems with high latency networks.
-\begin{figure}[t]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\caption{Various grid configurations with two networks parameters: $N1$ vs. $N2$}
\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
+\RCE{ok}
\label{fig:02}
\end{figure}
-\subsubsection{Network latency impacts on performance\\}
-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 Poison problem of size $150^3$. According to the results, a degradation of the network latency from $8\times 10^{-6}$ to $6\times 10^{-5}$ 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.
+\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.
-\begin{figure}[t]
+\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impacts on execution times}
+\caption{Network latency impacts on performances}
\label{fig:03}
\end{figure}
+\subsubsection{Network bandwidth impacts on performances\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
+$2\times16$ 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 $1$Gbs to $10$Gbs show the performances improvement for
+both algorithms by reducing the execution times. However, the Krylov two-stage
+algorithm presents a better performance gain 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}
-
-
-\begin{figure} [htbp]
+\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}). As shown in Figure~\ref{fig:05}, the execution
+times for both algorithms increase with increased matrix sizes. For all problem
+sizes, the 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}[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 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.
-\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}
-
-\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 ratio 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}
+\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.
