% Extension pour les liens intra-documents (tagged PDF)
% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}
+\usepackage{multirow}
+
\usepackage{url}
\DeclareUrlCommand\email{\urlstyle{same}}
A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
\label{eq:03}
\end{equation}
-where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES ({\it Generalized Minimal RESidual})~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
+where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
-\begin{figure}[t]
+\begin{figure}[htpb]
%\begin{algorithm}[t]
%\caption{Block Jacobi two-stage multisplitting method}
\begin{algorithmic}[1]
\end{equation}
The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
-\begin{figure}[t]
+\begin{figure}[htbp]
%\begin{algorithm}[t]
%\caption{Krylov two-stage method using block Jacobi multisplitting}
\begin{algorithmic}[1]
\subsection{Simulation of the two-stage methods using SimGrid toolkit}
\label{sec:04.02}
-One of our objectives when simulating the application in Simgrid is, as in real
+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 to ensure the
test reproducibility under the same conditions. According to our experience,
-very few modifications are required to adapt a MPI program for the Simgrid
+very few modifications are required to adapt a MPI program for the SimGrid
simulator using SMPI (Simulator MPI). The first modification is to include SMPI
-libraries and related header files (smpi.h). The second modification is to
+libraries and related header files (\verb+smpi.h+). The second modification is to
suppress all global variables by replacing them with local variables or using a
-Simgrid selector called "runtime automatic switching"
+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 and 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.
-\paragraph{Simgrid Simulator parameters}
-\ \\ \noindent Before running a Simgrid benchmark, many parameters for the
+\paragraph{Parameters of the simulation in SimGrid}
+\ \\ \noindent Before running a SimGrid benchmark, many parameters for the
computation platform must be defined. For our experiments, we consider platforms
in which several clusters are geographically distant, so there are intra and
inter-cluster communications. In the following, these parameters are described:
\begin{itemize}
- \item hostfile: hosts description file.
+ \item hostfile: hosts description file,
\item platform: file describing the platform architecture: clusters (CPU power,
-\dots{}), intra cluster network description, inter cluster network (bandwidth bw,
-latency lat, \dots{}).
+\dots{}), intra cluster network description, inter cluster network (bandwidth $bw$,
+latency $lat$, \dots{}),
\item archi : grid computational description (number of clusters, number of
-nodes/processors for each cluster).
+nodes/processors in each cluster).
\end{itemize}
\noindent
In addition, the following arguments are given to the programs at runtime:
\begin{itemize}
\item maximum number of inner iterations $\MIG$ and outer iterations $\MIM$,
\item inner precision $\TOLG$ and outer precision $\TOLM$,
- \item matrix sizes of the 3D Poisson problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively,
- \item matrix diagonal value is fixed to $6.0$ for synchronous Krylov multisplitting experiments and $6.2$ for asynchronous block Jacobi experiments,
+ \item matrix sizes of the problem: N$_{x}$, N$_{y}$ and N$_{z}$ on axis $x$, $y$ and $z$ respectively (in our experiments, we solve 3D problem, see Section~\ref{3dpoisson}),
+ \item matrix diagonal value is fixed to $6.0$ for synchronous experiments and $6.2$ for asynchronous ones,
\item matrix off-diagonal value is fixed to $-1.0$,
\item number of vectors in matrix $S$ (i.e. value of $s$),
\item maximum number of iterations $\MIC$ and precision $\TOLC$ for CGLS method,
\item execution mode: synchronous or asynchronous.
\end{itemize}
-It should also be noticed that both solvers have been executed with the Simgrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
+It should also be noticed that both solvers have been executed with the SimGrid selector \texttt{-cfg=smpi/running\_power} which determines the computational power (here 19GFlops) of the simulator host machine.
%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Experimental Results}
+\section{Experimental results}
\label{sec:expe}
-In this section, experiments for both Multisplitting algorithms are reported. First the 3D Poisson 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{The 3D Poisson problem}
-
-
+\label{3dpoisson}
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}
\frac{\partial^2}{\partial x^2}\phi(x,y,z)+\frac{\partial^2}{\partial y^2}\phi(x,y,z)+\frac{\partial^2}{\partial z^2}\phi(x,y,z)=f(x,y,z)\mbox{~in the domain~}\Omega
\textbf{Step 2}: Collect the software materials needed for the experimentation.
In our case, we have two variants algorithms for the resolution of the
-3D-Poisson problem: (1) using the classical GMRES; (2) and the Multisplitting
-method. In addition, the Simgrid simulator has been chosen to simulate the
-behaviors of the distributed applications. Simgrid is running in a virtual
+3D-Poisson problem: (1) using the classical GMRES; (2) and the multisplitting
+method. In addition, the SimGrid simulator has been chosen to simulate the
+behaviors of the distributed applications. SimGrid is running in a virtual
machine on a simple laptop. \\
\textbf{Step 3}: Fix the criteria which will be used for the future
on the one hand the algorithm execution mode (synchronous and asynchronous)
and on the other hand the execution time and the number of iterations to reach the convergence. \\
-\textbf{Step 4 }: Set up the different grid testbed environments that will be
-simulated in the simulator tool to run the program. The following architecture
-has been configured in Simgrid : 2x16, 4x8, 4x16, 8x8 and 2x50. The first number
+\textbf{Step 4}: Set up the different grid testbed environments that will be
+simulated in the simulator tool to run the program. The following architectures
+have been configured in SimGrid : 2$\times$16, 4$\times$8, 4$\times$16, 8$\times$8 and 2$\times$50. The first number
represents the number of clusters in the grid and the second number represents
-the number of hosts (processors/cores) in each cluster. The network has been
-designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
-latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
-(resp. inter-clusters backbone links). \\
+the number of hosts (processors/cores) in each cluster. \\
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
\textbf{Step 6} : Collect and analyze the output results.
-\subsection{Factors impacting distributed applications performance in
-a grid environment}
+\subsection{Factors impacting distributed applications performance in a grid environment}
When running a distributed application in a computational grid, many factors may
have a strong impact on the performance. First of all, the architecture of the
is the network configuration. Two main network parameters can modify drastically
the program output results:
\begin{enumerate}
-\item the network bandwidth (bw=bits/s) also known as "the data-carrying
+\item the network bandwidth ($bw$ in bits/s) also known as "the data-carrying
capacity" of the network is defined as the maximum of data that can transit
from one point to another in a unit of time.
-\item the network latency (lat : microsecond) defined as the delay from the
+\item the network latency ($lat$ in microseconds) defined as the delay from the
start time to send a simple data from a source to a destination.
\end{enumerate}
Upon the network characteristics, another impacting factor is the volume of data exchanged between the nodes in the cluster
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 bandwith and very low latency. In opposite, the inter-network connects
- clusters sometime via heterogeneous networks components throuth internet with
+ 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 of GMRES and Krylov Multisplitting algorithms in synchronous mode}
-In the scope of this paper, our first objective is to analyze when the Krylov
-Multisplitting 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.
-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.
+\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.
-The following paragraphs present the test conditions, the output results
-and our comments.\\
-
-
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
-\ \\
-% environment
+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}{r c }
- \hline
- Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =150 $\times$ 150 $\times$ 150\\ %\hline
- - & N$_{x}$ $\times$ N$_{y}$ $\times$ N$_{z}$ =170 $\times$ 170 $\times$ 170 \\ \hline
- \end{tabular}
-\caption{Test conditions: various grid configurations with the input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas défini..}\RC{Nx est défini, Ny? Nz?}
-\AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}. Idem dans le texte, les figures, etc.}}
+\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, $lat=8\mu$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}
+\caption{Parameters for the different simulations}
\label{tab:01}
\end{center}
\end{table}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}
+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).
-
-In this section, we analyze the performance of algorithms running on various
-grid configurations (2x16, 4x8, 4x16 and 8x8). First, the results in Figure~\ref{fig:01}
-show for all grid configurations 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{figure} [ht!]
- \begin{center}
- \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
- \end{center}
- \caption{Various grid configurations with the input matrix size $N_{x}=150$ and $N_{x}=170$\RC{idem}
-\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}}
- \label{fig:01}
-\end{figure}
-
-
-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. $48\%$) when running from 2x16=32 to 8x8=64 processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
-
-\subsubsection{Running on two different inter-clusters network speeds \\}
-
-\begin{table} [ht!]
+\begin{figure}[ht]
\begin{center}
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
- - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
- \end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
-\label{tab:02}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
\end{center}
-\end{table}
-
-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). \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 4x16 or 8x8: the reduction is about $2$. 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\%.
-
-
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
+\end{figure}
-%\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\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 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{Grid 2x16 and 4x8 with networks N1 vs N2
-\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}
\label{fig:02}
\end{figure}
-%\end{wrapfigure}
-
-
-\subsubsection{Network latency impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\hline
- 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}
-
+\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} [ht!]
+\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}
+\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.
-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. In addition, it appears that the
-Krylov multisplitting method tolerates more the network latency variation with a
-less rate increase of the execution time.\RC{Les 2 précédentes phrases me
- semblent en contradiction....} 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 $8.10^{-6}$.
-
-\subsubsection{Network bandwidth impacts on performance}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- 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}}
-\label{tab:04}
-\end{table}
-
-
-\begin{figure} [ht!]
+\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.}
+\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 & 4x8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
- Input matrix size & $N_{x}$ = From 40 to 200\\ \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} [ht!]
+\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 $N_{x} = N_{y}
-= N_{z} = 40$ 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 ($10$ times) of the number of iterations needed to
- reach the convergence for the classical GMRES algorithm when the matrix size
- go beyond $N_{x}=150$; \RC{C'est toujours pas clair... ok le nommbre d'itérations est 10 fois plus long mais la suite de la phrase ne veut rien dire}
-\item the classical GMRES execution time is almost the double for $N_{x}=140$
- compared with the Krylov multisplitting method.
-\end{enumerate}
-
-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 impacts on performance}
-
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid architecture & 2x16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \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}
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}
-In this section, 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.
+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
+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 version is faster than GMRES.
-The test conditions are summarized in the table~\ref{tab:07}: \\
-\begin{table} [ht!]
+\begin{table} [htbp]
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\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$^{-2}$\\
power (GFlops)
& 1 & 1 & 1 & 1.5 & 1.5 & 1.5 & 1.5 & 1 & 1.5 & 1.5 \\
\hline
- size (N)
+ size ($N^3$)
& 62 & 62 & 62 & 100 & 100 & 110 & 120 & 130 & 140 & 150 \\
\hline
Precision
