Modifs section 5.4.5

[rce2015.git] / paper.tex

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@@ -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} & $N1$: $bw$=1Gbs, $lat$=5$\times$10$^{-5}$ \\ 
-                                        & $N2$: $bw$=10Gbs, $lat$=8$\times$10$^{-6}$ \\
+\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}


@@ -549,201 +549,103 @@ 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}
-\  \\
-% 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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+\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}[ht]
+\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\\}
+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}[ht]
+\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}

 
 

+\subsubsection{Network latency impacts on performances\\}
+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}[ht]
+\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 performances\\}
+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}[ht]
+\centering
+\includegraphics[width=100mm]{network_bandwith_impact_on_execution_time.pdf}
+\caption{Network bandwith impacts on execution time}
+\label{fig:04}
+\end{figure}

 
 

+\subsubsection{Matrix size impacts on performances\\}
+In these experiments, the matrix size of the 3D Poisson problem is varied from $50^3$ to $190^3$ elements. The simulated computational grid is composed of 4 clusters of 8 processors each interconnected by the network $N2$ (see Table~\ref{tab:01}). Obviously, as shown in Figure~\ref{fig:05}, the execution times for both algorithms increase with increased matrix sizes.  For all problem sizes, GMRES algorithm is always slower than the Krylov two-stage algorithm. Moreover, for this benchmark, it seems that the greater the problem size is, the bigger the ratio between execution times of both algorithms is. We can also observe that for some problem sizes, the convergence (and thus the execution time) of the Krylov two-stage algorithm varies quite a lot. %This is due to the 3D partitioning of the 3D matrix of the Poisson problem. 
+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. 

 
 

-\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}
+\begin{figure}[ht]
+\centering
+\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
+\caption{Problem size impacts on execution times}
+\label{fig:05}

 \end{figure}
 
 
 
 \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}
-\ \\
-\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 $6.10^{-5}$.

 
 
 
 

-\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}
-\ \\
-\begin{table} [ht!]
-\centering
-\begin{tabular}{r c }
- \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 50$^{3}$ to 190$^{3}$\\ \hline
- \end{tabular}
-\caption{Test conditions: Input matrix size impacts}
-\label{tab:05}
-\end{table}

 
 
 
 

-\begin{figure} [htbp]
-\centering
-\includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
-\caption{Problem size impacts on execution time}
-\label{fig:05}
-\end{figure}

 
 

-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
-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.
+\subsubsection{CPU Power impacts on performance\\}

 
 

-\subsubsection{CPU Power impacts on performance}

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


@@ -804,7 +706,7 @@ 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.