Modifs section 5.4.3

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
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--- a/paper.tex
+++ b/paper.tex
@@ -569,19 +569,19 @@ The execution times between both algorithms is significant with different grid a
 \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$^3$}
 \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$^3$}

-\LZK{CE, la légende de la Figure 3 est trop large. Remplacer les N$_x\times$N$_y\times$N$_z$ par $Mat1$=150$^3$ et $Mat2$=170$^3$ comme dans la Table 1}

 \label{fig:01}
 \end{figure}
 
 \subsubsection{Simulations for two different inter-clusters network speeds\\}
 \label{fig:01}
 \end{figure}
 
 \subsubsection{Simulations for two different inter-clusters network speeds\\}

-In Figure~\ref{fig:02} we present the execution times of both algorithms to solve a 3D Poisson problem of size $150^3$ on two different simulated network $N1$ and $N2$ (see Table~\ref{tab:01}). As it was previously said, we can see from the figure that the Krylov two-stage algorithm is more sensitive to the number of clusters than the GMRES algorithm. However, we can notice an interesting behavior of the Krylov two-stage algorithm. It is less sensitive to bad network bandwidth and latency for the inter-clusters links than the GMRES algorithms. This means that the multisplitting methods are more efficient for distributed systems with high latency networks.
-
-%% 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\%.
+In  Figure~\ref{fig:02} we  present the  execution times  of both  algorithms to
+solve a  3D Poisson problem of  size $150^3$ on two  different simulated network
+$N1$ and $N2$ (see Table~\ref{tab:01}). As previously mentioned, we can see from
+this figure  that the Krylov two-stage  algorithm is sensitive to  the number of
+clusters (i.e. it is better to have a small number of clusters). However, we can
+notice an  interesting behavior of  the Krylov  two-stage algorithm. It  is less
+sensitive to bad network bandwidth and latency for the inter-clusters links than
+the  GMRES algorithms.  This  means  that the  multisplitting  methods are  more
+efficient for distributed systems with high latency networks.

 
 \begin{figure}[t]
 \centering
 
 \begin{figure}[t]
 \centering


@@ -591,7 +591,15 @@ In Figure~\ref{fig:02} we present the execution times of both algorithms to solv
 \label{fig:02}
 \end{figure}
 
 \label{fig:02}
 \end{figure}
 

+\subsubsection{Network latency impacts on performance\\}
+Figure~\ref{fig:03} shows the impact of the network latency on the performances of both algorithms. The simulation is conducted on a computational grid of 2 clusters of 16 processors each (i.e. configuration 2$\times$16) interconnected by a network of bandwidth $bw$=1Gbs to solve a 3D Poison problem of size $150^3$. According to the results, a degradation of the network latency from $8\times 10^{-6}$ to $6\times 10^{-5}$ implies an absolute execution time increase for both algorithms, but not with the same rate of degradation. The GMRES algorithm is more sensitive to the latency degradation than the Krylov two-stage algorithm. 

 
 

+\begin{figure}[t]
+\centering
+\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
+\caption{Network latency impacts on execution times}
+\label{fig:03}
+\end{figure}

 
 
 
 
 
 


@@ -625,36 +633,8 @@ In Figure~\ref{fig:02} we present the execution times of both algorithms to solv
 
 
 
 
 
 

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

 
 

-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2 $\times$ 16\\ %\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} [htbp]
-\centering
-\includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
-\caption{Network latency impacts on execution time}
-%\AG{\np{E-6}}}
-\label{fig:03}
-\end{figure}

 
 

-In Table~\ref{tab:03}, parameters  for the influence of the  network latency are
-reported.  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. 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 $6.10^{-5}$ second.

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


@@ -789,7 +769,7 @@ benchmarks have  been performed with  various combination of the  grid resources
 in  Table~\ref{tab:07}. In  order to  compare  the execution  times, this  table
 reports the  relative gain between both  algorithms. It is defined  by the ratio
 between  the   execution  time  of   GMRES  and   the  execution  time   of  the
 in  Table~\ref{tab:07}. In  order to  compare  the execution  times, this  table
 reports the  relative gain between both  algorithms. It is defined  by the ratio
 between  the   execution  time  of   GMRES  and   the  execution  time   of  the

-multisplitting.  The  ration  is  greater  than  one  because  the  asynchronous
+multisplitting.  The  ratio  is  greater  than  one  because  the  asynchronous

 multisplitting version is faster than GMRES.
 
 
 multisplitting version is faster than GMRES.