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%\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?}
\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.
+\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
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ Network & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-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 matrix sizes 150$^3$ or 170$^3$}
+\LZK{Ce sont les caractéristiques du réseau intra ou inter clusters? Ce n'est pas précisé...}
+=======
+ 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
\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?}
+\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.}}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:01}
\end{center}
\end{table}
+<<<<<<< HEAD
+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...}
+=======
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}
+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...}
-
+%\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...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\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}
-
+<<<<<<< HEAD
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?}
+=======
+
+Secondly, 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
+(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?}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
-\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
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{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 configurations 2$\times$16 and 4$\times$8 with networks N1 vs. N2}
+=======
+ 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
\end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with networks N1 vs N2}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
\label{tab:02}
\end{center}
\end{table}
+<<<<<<< HEAD
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...}
+=======
+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...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
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 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\%.
\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{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2
\AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
\label{fig:02}
\end{figure}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\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}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\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}
\centering
\begin{tabular}{r c }
\hline
- Grid Architecture & 4x8\\ %\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
\end{tabular}
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
+ 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
\end{tabular}
\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$^{-2}$\\
