X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/3b175ad474db1b6208b1115e91d0b5bfde6f0288..538afc25a7a90630d2f90891d0a4d700bfe3460f:/paper.tex

diff --git a/paper.tex b/paper.tex
index 60fcffc..a2e23a2 100644
--- a/paper.tex
+++ b/paper.tex
@@ -530,49 +530,55 @@ and  between distant  clusters.  This parameter is application dependent.
 
 
 \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. In what follows, we will present the test conditions, the output results and our comments. For all simulations, we fix the network parameters of the intra-cluster links: the bandwidth $bw$=10Gbs and the latency $lat$=8$\times$10$^{-6}$.
+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.
 
-\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
-\  \\
-% environment
-
- The network of intra-clusters links has been
-designed to  operate with a bandwidth  equals to 10Gbits and a latency of 8$\times$10$^{-6}$ seconds. \\
-
-\RC{Je ne comprends plus rien CE : pourquoi dans 5.4.1 il y a 2 network et aussi dans 5.4.2. Quelle est la diffÃ©rence? Dans la figure 3 de la section 5.4.1 pourquoi il n'y a pas N1 et N2?}
+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$\times$10$^{-6}$. In what follows, we will present the test conditions, the output results and our comments. 
 
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{ll }
- \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 matrix sizes 150$^3$ or 170$^3$}
-%\LZK{Ce sont les caractÃ©ristiques du rÃ©seau intra ou inter clusters? Ce n'est pas prÃ©cisÃ©...}
-%\RCE{oui c est precise}
+\begin{tabular}{ll}
+\hline
+Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 
+                                        & $N2$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+\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}
+\  \\
+% environment
+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 $N1$ (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.
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-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 vÃ©rifie...}
-%\RC{Les lÃ©gendes ne sont pas explicites...}
-%\RCE{Corrige}
 
 \begin{figure} [htbp]
   \begin{center}
@@ -708,7 +714,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
  \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
+ Input matrix size & $N_{x} \times N_{y} \times N_{z}$ = From 50$^{3}$ to 190$^{3}$\\ \hline
  \end{tabular}
 \caption{Test conditions: Input matrix size impacts}
 \label{tab:05}
@@ -722,20 +728,15 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \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 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'itÃ©rations 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}
+In  these  experiments, the  input  matrix  size has  been  set  from $50^3$  to
+$190^3$. Obviously, as shown in Figure~\ref{fig:05}, the execution time for both
+algorithms increases when the input matrix size also increases.  For all problem
+sizes, GMRES is always slower than the Krylov multisplitting. Moreover, for this
+benchmark, it seems that  the greater the problem size is,  the bigger the ratio
+between both  algorithm execution  times is.  We can also  observ that  for some
+problem   sizes,  the   Krylov   multisplitting  convergence   varies  quite   a
+lot. Consequently the execution times in that cases also varies.
+
 
 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
