% 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.
+\LZK{Pas du tout claire la dernière phrase (For a systematic...)!!}
-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 & 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
+ Network & N1 : $bw$=1Gbits/s, $lat$=5.10$^{-5}$ \\ %\hline
+ \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...}
\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.
-
+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...}
-
\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?}
\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
+and 4x8). We can observe 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?}
+$40\%$ 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?}
+\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?}
-\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}{*}{Network} & N1: $bw$=1Gbs, $lat$=5.10$^{-5}$ \\ %\hline
+ & N2: $bw$=10Gbs, $lat$=8.10$^{-6}$ \\
+ 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...}
+slow inter-cluster network (N1) and also on a more 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\%.
