% 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}}
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?}
\textbf{Step 5}: Conduct an extensive and comprehensive testings
within these configurations by varying the key parameters, especially
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.
+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}
-The following paragraphs present the test conditions, the output results
-and our comments.\\
+In what follows, we will 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}
+%\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 & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
- Inter 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 matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 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 (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. 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}. 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}
+ \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}
\label{fig:01}
\end{figure}
-
-Secondly, the execution times between the two algorithms is significant with different
+The execution times between the two algorithms is significant with different
grid architectures, even with the same number of processors (for example, 2 $\times$ 16
-and 4 $\times$ 8). We can observ the sensitivity of the Krylov multisplitting method
+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 reduction of the execution time for GMRES (resp. Multisplitting) algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas très clair, c'est pas précis de dire qu'un algo perform mieux qu'un autre, selon quel critère?}
+in the grid: in average, the GMRES (resp. Multisplitting) algorithm performs
+$40\%$ 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 & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
- Inter Networks & 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 2 $\times$ 16 and 4 $\times$ 8 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}
In this section, the experiments compare the behavior of the algorithms running on a
-speeder inter-cluster network (N1) and also on a less performant network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut définir cela avant...}
+speeder inter-cluster network (N2) and also on a less performant network (N1) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut définir cela avant...}
Figure~\ref{fig:02} shows that end users will reduce the execution time
-for both algorithms when using a grid architecture like 4 $\times$ 16 or 8 $\times$ 8: the reduction 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 2 $\times$ 16 and 4 $\times$ 8 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}
\begin{tabular}{r c }
\hline
Grid Architecture & 2 $\times$ 16\\ %\hline
- Network & N1 : bw=1Gbs \\ %\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}
\ \\
\begin{tabular}{r c }
\hline
Grid Architecture & 2 $\times$ 16\\ %\hline
- Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\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}
\begin{tabular}{r c }
\hline
Grid Architecture & 4 $\times$ 8\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
- Input matrix size & $N_{x}$ = From 40 to 200\\ \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}
\begin{tabular}{r c }
\hline
Grid architecture & 2 $\times$ 16\\ %\hline
- Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
- Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \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
