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diff --git a/paper.tex b/paper.tex
index e37e28c49453eb076e2812bf27a072f18e22e2eb..ab8f9abfcbef3367f53947788140445861fcb16a 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -539,8 +539,8 @@ Table~\ref{tab:01} summarizes the parameters used in the different simulations:
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\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}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+                                        & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 

 \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}
 \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}


@@ -549,14 +549,44 @@ Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 a
 \end{center}
 \end{table}
 
 \end{center}
 \end{table}
 

+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes\\}

 
 

+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    $N2$    (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. In fact, with multisplitting  algorithms, the number of splitting (in
+our case, it is the number of clusters) influences on the convergence speed. The
+higher the number  of splitting is, the slower the  convergence of the algorithm
+is (see the output results obtained from configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs. 8$\times$8).
+
+The execution times between both algorithms is significant with different grid architectures. The synchronous Krylov two-stage algorithm presents better performances than the GMRES algorithm, even for a high number of clusters (about $32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can observe a better sensitivity of the Krylov two-stage algorithm (compared to the GMRES one) when scaling up the number of the processors in the computational grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is about $40\%$ better on 64 processors (grid of 8$\times$8) than 32 processors (grid of 2$\times$16). 
+
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
+\end{center}
+\caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$}
+\label{fig:01}
+\end{figure}

 
 

+\subsubsection{Simulations for two different inter-clusters network speeds\\}

 
 

-\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
-\  \\
-% environment
+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  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\%.

 
 

-Table~\ref{tab:01} summarizes the different parameters used in the simulations: the grid architectures, the network of inter-cluster backbone links and the matrix sizes of the 3D Poisson problem.
+\begin{figure}[t]
+\centering
+\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
+\caption{Various grid configurations with networks $N1$ vs. $N2$}
+\label{fig:02}
+\end{figure}

 
 
 
 
 
 


@@ -568,84 +598,19 @@ Table~\ref{tab:01} summarizes the different parameters used in the simulations:
 
 
 
 
 
 

-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} [htbp]
-  \begin{center}
-    \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
-  \end{center}
-  \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, 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 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{Simulations for two different inter-clusters network speeds \\}

 
 

-\begin{table} [ht!]
-\begin{center}
-\begin{tabular}{ll}
- \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 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 (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 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{wrapfigure}{l}{100mm}
-\begin{figure} [htbp]
-\centering
-\includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\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}

 
 

+\subsubsection{Network latency impacts on performance\\}

 
 

-\subsubsection{Network latency impacts on performance}
-\ \\

 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }
 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }


@@ -673,11 +638,11 @@ 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. The  execution  time  factor
 between the two algorithms  varies from 2.2 to 1.5 times  with a network latency
 increase of more than $75\%$ (resp.   $82\%$) of the execution for the classical
 GMRES  (resp.   Krylov  multisplitting)  algorithm. 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 $6.10^{-5}$.
+decreasing from $8.10^{-6}$ to $6.10^{-5}$ second.

 
 
 
 

-\subsubsection{Network bandwidth impacts on performance}
-\ \\
+\subsubsection{Network bandwidth impacts on performance\\}
+

 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }
 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }


@@ -709,8 +674,8 @@ Figure~\ref{fig:04}). However,  in this  case, the Krylov  multisplitting method
 presents a better  performance in the considered bandwidth interval  with a gain
 of $40\%$ which is only around $24\%$ for the classical GMRES.
 
 presents a better  performance in the considered bandwidth interval  with a gain
 of $40\%$ which is only around $24\%$ for the classical GMRES.
 

-\subsubsection{Input matrix size impacts on performance}
-\ \\
+\subsubsection{Input matrix size impacts on performance\\}
+

 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }
 \begin{table} [ht!]
 \centering
 \begin{tabular}{r c }


@@ -744,9 +709,10 @@ 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
 size scale up.  It  should be noticed that the same test has  been done with the
 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 2 $\times$ 16 leading to the same conclusion.
+grid 4 $\times$ 8 leading to the same conclusion.
+
+\subsubsection{CPU Power impacts on performance\\}

 
 

-\subsubsection{CPU Power impacts on performance}

 
 \begin{table} [htbp]
 \centering
 
 \begin{table} [htbp]
 \centering