Modifs section 5.4.4

[rce2015.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index ab8f9abfcbef3367f53947788140445861fcb16a..ca1749c8a82b6573e2ed70ecb5bb678e5f687da8 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -532,15 +532,15 @@ 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.
 
 \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.
 

-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. 
+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 $lat=8\mu$s. 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\\ 
 
 \begin{table} [ht!]
 \begin{center}
 \begin{tabular}{ll}
 \hline
 Grid architecture                       & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ 

-\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
-                                        & $N2$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 
+\multirow{2}{*}{Network inter-clusters} & $N1$: $bw$=10Gbs, $lat=8\mu$s \\
+                                        & $N2$: $bw$=1Gbs, $lat=50\mu$s \\ 

 \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}


@@ -573,21 +573,51 @@ The execution times between both algorithms is significant with different grid a
 \end{figure}
 
 \subsubsection{Simulations for two different inter-clusters network speeds\\}
 \end{figure}
 
 \subsubsection{Simulations for two different inter-clusters network speeds\\}

-
-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
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Various grid configurations with networks $N1$ vs. $N2$}
 
 \begin{figure}[t]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Various grid configurations with networks $N1$ vs. $N2$}

+\LZK{CE, remplacer les ``,'' des décimales par un ``.''}

 \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 Poisson problem of size $150^3$. According to the results, a degradation of the network latency from $8\mu$s to $60\mu$s 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}
+
+\subsubsection{Network bandwidth impacts on performance\\}
+Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of 2$\times$16 processors interconnected by a network of latency $lat=50\mu$s to solve a 3D Poisson problem of size $150^3$. The results of increasing the network bandwidth from 1Gbs to 10Gbs show the performances improvement for both algorithms by reducing the execution times. However, the Krylov two-stage algorithm presents a better performance in the considered bandwidth interval with a gain of $40\%$ compared to only about $24\%$ for the classical GMRES algorithm.
+
+\begin{figure}[t]
+\centering
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impacts on execution time}
+\label{fig:04}
+\end{figure}
+
+
+
+
+
+
+
+

 
 
 
 
 
 


@@ -609,70 +639,15 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 
 
 
 
 
 

-\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\\}

 
 

-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \hline
- Grid Architecture & 2 $\times$ 16\\ %\hline
-\multirow{2}{*}{Inter Network N1} & $bw$=From 1Gbs to 10 Gbs \\ %\hline
-                          & $lat$= 5.10$^{-5}$ second \\
- Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
- \end{tabular}
-\caption{Test conditions: Network bandwidth impacts}
-%  \RC{Qu'est ce qui varie ici? Il n'y a pas de variation dans le tableau}
-%\RCE{C est le bw}
-\label{tab:04}
-\end{table}

 
 
 
 

-\begin{figure} [htbp]
-\centering
-\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
-\caption{Network bandwith impacts on execution time}
-%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
-%\RCE{Corrige}
-\label{fig:04}
-\end{figure}

 
 

-The results  of increasing  the network  bandwidth show  the improvement  of the
-performance  for   both  algorithms   by  reducing   the  execution   time  (see
-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.

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


@@ -773,7 +748,7 @@ 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.