\end{abstract}
%\keywords{Algorithm; distributed; iterative; asynchronous; simulation; simgrid;
-%performance}
+%performance}
\keywords{ Performance evaluation, Simulation, SimGrid, Synchronous and asynchronous iterations, Multisplitting algorithms}
\maketitle
given application for a given architecture. In this way and in order to reduce
the access cost to these computing resources it seems very interesting to use a
simulation environment. The advantages are numerous: development life cycle,
-code debugging, ability to obtain results quickly,~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
+code debugging, ability to obtain results quickly~\ldots. In counterpart, the simulation results need to be consistent with the real ones.
In this paper we focus on a class of highly efficient parallel algorithms called
\emph{iterative algorithms}. The parallel scheme of iterative methods is quite
simple. It generally involves the division of the problem into several
\emph{blocks} that will be solved in parallel on multiple processing
-units. Each processing unit has to compute an iteration, to send/receive some
+units. Each processing unit has to compute an iteration to send/receive some
data dependencies to/from its neighbors and to iterate this process until the
-convergence of the method. Several well-known methods demonstrate the
+convergence of the method. Several well-known studies demonstrate the
convergence of these algorithms~\cite{BT89,bahi07}. In this processing mode a
task cannot begin a new iteration while it has not received data dependencies
-from its neighbors. We say that the iteration computation follows a synchronous
-scheme. In the asynchronous scheme a task can compute a new iteration without
-having to wait for the data dependencies coming from its neighbors. Both
-communication and computations are asynchronous inducing that there is no more
-idle time, due to synchronizations, between two iterations~\cite{bcvc06:ij}.
-This model presents some advantages and drawbacks that we detail in
-section~\ref{sec:asynchro} but even if the number of iterations required to
-converge is generally greater than for the synchronous case, it appears that
-the asynchronous iterative scheme can significantly reduce overall execution
-times by suppressing idle times due to synchronizations~(see~\cite{bahi07}
-for more details).
+from its neighbors. We say that the iteration computation follows a
+\textit{synchronous} scheme. In the asynchronous scheme a task can compute a new
+iteration without having to wait for the data dependencies coming from its
+neighbors. Both communication and computations are \textit{asynchronous}
+inducing that there is no more idle time, due to synchronizations, between two
+iterations~\cite{bcvc06:ij}. This model presents some advantages and drawbacks
+that we detail in section~\ref{sec:asynchro} but even if the number of
+iterations required to converge is generally greater than for the synchronous
+case, it appears that the asynchronous iterative scheme can significantly
+reduce overall execution times by suppressing idle times due to
+synchronizations~(see~\cite{bahi07} for more details).
Nevertheless, in both cases (synchronous or asynchronous) it is very time
consuming to find optimal configuration and deployment requirements for a given
k\geq\MIM\mbox{~or~}\|x_\ell^{k+1}-x_\ell^k\|_{\infty }\leq\TOLM,
\label{eq:04}
\end{equation}
-where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
+where $\MIM$ is the maximum number of outer iterations and $\TOLM$ is the tolerance threshold for the two-stage algorithm.
The second two-stage algorithm is based on synchronous outer iterations. We propose to use the Krylov iteration based on residual minimization to improve the slow convergence of the multisplitting methods. In this case, a $n\times s$ matrix $S$ is set using solutions issued from the inner iteration
\begin{equation}
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
-Simgrid to simulate the grid environment. \RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}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.
+Simgrid to simulate the grid environment.
+
+%\RC{On vire cette phrase ?} \RCE {Si c'est la phrase d'avant sur les threads, je pense qu'on peut la retenir car c'est l'explication du pourquoi Simgrid n'aime pas les variables globales. Si c'est pas bien dit, on peut la reformuler. Si c'est la phrase ci-apres, effectivement, on peut la virer si elle preterais a discussion}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}
\item matrix diagonal value = 6.0 for synchronous Krylov multisplitting experiments and 6.2 for asynchronous block Jacobi experiments; \RC{CE tu vérifies, je dis ca de tête}
\item matrix off-diagonal value;
\item execution mode: synchronous or asynchronous;
- \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous}
+ \RCE {C'est ok la liste des arguments du programme mais si Lilia ou toi pouvez preciser pour les arguments pour CGLS ci dessous} \RC{Vu que tu n'as pas fait varier ce paramètre, on peut ne pas en parler}
\item Size of matrix S;
- \item Maximum number of iterations and tolerance threshold for CGLS.
+ \item Maximum number of iterations and tolerance threshold for CGLS.
\end{itemize}
It should also be noticed that both solvers have been executed with the Simgrid selector -cfg=smpi/running\_power which determines the computational power (here 19GFlops) of the simulator host machine.
\section{Experimental Results}
\label{sec:expe}
-In this section, experiments for both Multisplitting algorithms are reported. First the problem used in our experiments is described.
+In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson problem used in our experiments is described.
+
+\subsection{3D Poisson}
+
We use our two-stage algorithms to solve the well-known Poisson problem $\nabla^2\phi=f$~\cite{Polyanin01}. In three-dimensional Cartesian coordinates in $\mathbb{R}^3$, the problem takes the following form
\begin{equation}
\begin{equation*}
\phi(x,y,z)=0\mbox{~on the boundary~}\partial\Omega
\end{equation*}
-where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
+where the real-valued function $\phi(x,y,z)$ is the solution sought, $f(x,y,z)$ is a known function and $\Omega=[0,1]^3$. The 3D discretization of the Laplace operator $\nabla^2$ with the finite difference scheme includes 7 points stencil on the computational grid. The numerical approximation of the Poisson problem on three-dimensional grid is repeatedly computed as $\phi=\phi^\star$ such that
\begin{equation}
\begin{array}{ll}
\phi^\star(x,y,z)=&\frac{1}{6}(\phi(x-h,y,z)+\phi(x,y-h,z)+\phi(x,y,z-h)\\&+\phi(x+h,y,z)+\phi(x,y+h,z)+\phi(x,y,z+h)\\&-h^2f(x,y,z))
\end{array}
\label{eq:08}
\end{equation}
-until convergence where $h$ is the grid spacing between two adjacent elements in the 3D computational grid.
+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}
speed. The network between distant clusters might be a bottleneck for the
global performance of the application.
-\subsection{Comparing GMRES and Multisplitting algorithms in
-synchronous mode}
+\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
-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
-\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
-simulator will confirm the targeted outcomes, particularly for poor and
-slow networks, focusing on the impact on the communication performance
-on the chosen class of algorithm.
+In the scope of this paper, our first objective is to analyze when the Krylov
+Multisplitting method has better performances than the classical GMRES
+method. With an iterative method, better performances mean a smaller number of
+iterations and execution time before reaching the convergence. For a systematic
+study, the experiments should figure out that, for various grid parameters
+values, the simulator will confirm the targeted outcomes, particularly for poor
+and slow networks, focusing on the impact on the communication performance on
+the chosen class of algorithm.
The following paragraphs present the test conditions, the output results
and our comments.\\
-\textit{3.a Executing the algorithms on various computational grid
+\subsubsection{Execution of the the algorithms on various computational grid
architecture and scaling up the input matrix size}
-\\
-
+\ \\
% environment
-\begin{footnotesize}
+
+\begin{figure} [ht!]
+\begin{center}
\begin{tabular}{r c }
\hline
Grid & 2x16, 4x8, 4x16 and 8x8\\ %\hline
Input matrix size & N$_{x}$ x N$_{y}$ x N$_{z}$ =150 x 150 x 150\\ %\hline
- & N$_{x}$ x N$_{y}$ x N$_{z}$ =170 x 170 x 170 \\ \hline
\end{tabular}
-Table 1 : Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \\
+\caption{Clusters x Nodes with N$_{x}$=150 or N$_{x}$=170 \RC{je ne comprends pas la légende... Ca ne serait pas plutot Characteristics of cluster (mais il faudrait lui donner un nom)}}
+\end{center}
+\end{figure}
-\end{footnotesize}
%\RCE{J'ai voulu mettre les tableaux des données mais je pense que c'est inutile et ça va surcharger}
-In this section, we compare the algorithms performance running on various grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in figure 3 show for all grid configuration the non-variation of the number of iterations of classical GMRES for a given input matrix size; it is not
-the case for the multisplitting method.
+In this section, we analyze the performences of algorithms running on various
+grid configuration (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
+show for all grid configuration the non-variation of the number of iterations of
+classical GMRES for a given input matrix size; it is not the case for the
+multisplitting method.
+
+\RC{CE attention tu n'as pas mis de label dans tes figures, donc c'est le bordel, j'en mets mais vérifie...}
+\RC{Les légendes ne sont pas explicites...}
+
-%\begin{wrapfigure}{l}{100mm}
\begin{figure} [ht!]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-\caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
-%\label{overflow}}
+ \begin{center}
+ \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+ \end{center}
+ \caption{Cluster x Nodes N$_{x}$=150 and N$_{x}$=170}
+ \label{fig:01}
\end{figure}
-%\end{wrapfigure}
-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.
-% environment
-\begin{footnotesize}
+\subsubsection{Running on two different speed cluster inter-networks}
+\ \\
+
+\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$^{-6}$.
-
-\textit{\\3.d Network bandwidth impacts on performance\\}
+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$^{-6}$.
-% 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.
-
-\textit{\\3.e Input matrix size impacts on performance\\}
+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.
-% 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}.
+\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. 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
