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% et l'affichage correct des URL (commande \url{http://example.com})
%\usepackage{hyperref}
+\usepackage{multirow}
+
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\DeclareUrlCommand\email{\urlstyle{same}}
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-\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}
\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
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
+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$50. The first number
represents the number of clusters in the grid and the second number represents
-the number of hosts (processors/cores) in each cluster. The network has been
+the number of hosts (processors/cores) in each cluster. The network has been
designed to operate with a bandwidth equals to 10Gbits (resp. 1Gbits/s) and a
latency of 8.10$^{-6}$ seconds (resp. 5.10$^{-5}$) for the intra-clusters links
-(resp. inter-clusters backbone links). \\
+(resp. inter-clusters backbone links). \\
+
+\LZK{Il me semble que le bw et lat des deux réseaux varient dans les expés d'une simu à l'autre. On vire la dernière phrase?}
+\RC{il me semble qu'on peut laisser ca}
\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}
+\subsection{Comparison of GMRES and Krylov two-stage 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
+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.
-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 what follows, we will present the test conditions, the output results and our comments.
-The following paragraphs present the test conditions, the output results
-and our comments.\\
+%%RAPH : on vire ca, c'est pas clair et pas important
+%For a systematic study, the experiments should figure out that, for various
+%grid parameters values, the simulator will confirm Multisplitting method better performance compared to classical GMRES, particularly on poor and slow networks.
+%\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}
+%\RCE { Reformule autrement}
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
+
+%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
\ \\
% environment
\begin{table} [ht!]
\begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll }
\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
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ \multirow{2}{*}{Network} & Inter (N2): $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ & Intra (N1): $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ \multirow{2}{*}{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.}}
+\caption{Test conditions: various grid configurations with the matrix sizes 150$^3$ or 170$^3$}
+%\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+%\RCE{oui c est precise}
\label{tab:01}
\end{center}
\end{table}
-
-
-
-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...}
-
+In this section, we analyze the simulations conducted on various grid
+configurations presented in Table~\ref{tab:01}. It should be noticed that two
+networks are considered: N1 is the network between clusters (inter-cluster) and
+N2 is the network inside a cluster (intra-cluster). Figure~\ref{fig:01} shows,
+for all grid configurations and a given matrix size, a non-variation in the
+number of iterations for the classical GMRES algorithm, which is not the case of
+the Krylov two-stage algorithm.
+%% 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...}
+%\RCE{Corrige}
\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.}}
+ \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+%\AG{Utiliser le point comme séparateur décimal et non la virgule. Idem dans les autres figures.}
+%\LZK{Pour quelle taille du problème sont calculés les nombres d'itérations? Que représente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
+ %\RCE {Corrige}
+ \RC{Idéalement dans la légende il faudrait insiquer Pb size=$150^3$ ou $170^3$ car pour l'instant Nx=150 ca n'indique rien concernant Ny et Nz}
\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
+grid architectures, even with the same number of processors (for example, 2 $\times$ 16
+and 4 $\times 8$). We can observe a better 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?}
+$40\%$ better (resp. $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors/cores (grid 8 $\times$ 8). Note that even with a grid 8 $\times$ 8 having the maximum number of clusters, the execution time of the multisplitting method is in average 32\% less compared to GMRES.
+\RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exécution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+\RCE{Remarquez que meme avec une grille 8x8, le multi est toujours plus performant}
-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Simulations for two different inter-clusters network speeds \\}
\begin{table} [ht!]
\begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll}
\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
+ Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{Inter Network} & N1: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ & N2: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+ 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}
+\caption{Test conditions: grid configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
\label{tab:02}
\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...}
+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 4x16 or 8x8: the reduction is about $2$. The results depict also that when
+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\%.
\begin{figure} [ht!]
\centering
\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Various grid configurations with networks N1 vs N2
\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
+\RCE{Corrige}
\label{fig:02}
\end{figure}
%\end{wrapfigure}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\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} [ht!]
\centering
\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
\label{fig:03}
\end{figure}
-
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 $8.10^{-6}$.
+(resp. Krylov multisplitting) algorithm which means that the GMRES seems tolerate more the network latency variation with a less rate increase of the execution time. However, 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 $6.10^{-5}$.
+
+\RC{Les 2 précédentes phrases me semblent en contradiction....}
+\RCE{Reformule}
\subsubsection{Network bandwidth impacts on performance}
\ \\
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\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}
\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}
\label{fig:04}
\end{figure}
\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
+ 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 40$^{3}$ to 200$^{3}$\\ \hline
\end{tabular}
\caption{Test conditions: Input matrix size impacts}
\label{tab:05}
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
+ \item the important increase ($10$ times) of the number of iterations needed to
+ reach the convergence for the classical GMRES algorithm particularly, 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}
+ \RCE{Le nombre d'iterations augmente de 10 fois, cela surtout a partir de N=150}
+
\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.
+grid 2 $\times$ 16 leading to the same conclusion.
\subsubsection{CPU Power impacts on performance}
\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
+ 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}
performance.
\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
+\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain}
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
\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$^{-2}$\\
