X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/87062e3e1ac002d9b4c2175d7844aff13f17404e..010432af7f642984f9782a37ba4290313a250a2a:/paper.tex?ds=sidebyside

diff --git a/paper.tex b/paper.tex
index d5a458c..cb95155 100644
--- a/paper.tex
+++ b/paper.tex
@@ -24,6 +24,8 @@
 % Extension pour les liens intra-documents (tagged PDF)
 % et l'affichage correct des URL (commande \url{http://example.com})
 %\usepackage{hyperref}
+\usepackage{multirow}
+
 
 \usepackage{url}
 \DeclareUrlCommand\email{\urlstyle{same}}
@@ -490,7 +492,9 @@ represents the number  of clusters in the grid and  the second number represents
 the number  of hosts (processors/cores)  in each  cluster. The network has been
 designed to  operate with a bandwidth  equals to 10Gbits (resp.  1Gbits/s) and a
 latency of 8.10$^{-6}$ seconds (resp.  5.10$^{-5}$) for the intra-clusters links
-(resp.  inter-clusters backbone links). \\
+(resp.  inter-clusters backbone links).  \\
+
+\LZK{Il me semble que le bw et lat des deux rÃ©seaux varient dans les expÃ©s d'une simu Ã  l'autre. On vire la derniÃ¨re phrase?}
 
 \textbf{Step 5}: Conduct an extensive and comprehensive testings
 within these configurations by varying the key parameters, especially
@@ -531,92 +535,134 @@ and  between distant  clusters.  This parameter is application dependent.
  a lower speed.  The network  between distant  clusters might  be a  bottleneck
  for  the global performance of the application.
 
-\subsection{Comparison of GMRES and Krylov Multisplitting algorithms in synchronous mode}
+\subsection{Comparison of GMRES and Krylov two-stage algorithms in synchronous mode}
 
 In the scope  of this paper, our  first objective is to analyze  when the Krylov
-Multisplitting  method   has  better  performance  than   the  classical  GMRES
-method. With a synchronous  iterative method, better performance means a
+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.
 For a systematic study,  the experiments  should figure  out  that, for  various
 grid  parameters values, the simulator will confirm  the targeted outcomes,
 particularly for poor and slow  networks, focusing on the  impact on the
 communication  performance on the chosen class of algorithm.
+\LZK{Pas du tout claire la derniÃ¨re phrase (For a systematic...)!!}
 
-The following paragraphs present the test conditions, the output results
-and our comments.\\
+In what follows, we will present the test conditions, the output results and our comments.\\
 
-
-\subsubsection{Execution of the algorithms on various computational grid
-architectures and scaling up the input matrix size}
+%\subsubsection{Execution of the algorithms on various computational grid architectures and scaling up the input matrix size}
+\subsubsection{Simulations for various grid architectures and scaling-up matrix sizes}
 \ \\
 % environment
 
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll }
  \hline
- Grid Architecture & 2x16, 4x8, 4x16 and 8x8\\ %\hline
- Network & N2 : bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture & 2$\times$16, 4$\times$8, 4$\times$16 and 8$\times$8\\ %\hline
+ Network           & N1 : $bw$=1Gbits/s, $lat$=5$\times$10$^{-5}$ \\ %\hline
+ \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©...}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16 and 8 $\times$ 8\\ %\hline
+ Inter Network N2 & bw=1Gbits/s - lat=5.10$^{-5}$ \\ %\hline
  Input 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 input matix size N$_{x}$=150 or N$_{x}$=170 \RC{N2 n'est pas dÃ©fini..}\RC{Nx est dÃ©fini, Ny? Nz?}
+\caption{Test conditions: various grid configurations with the input matrix size N$_{x}$=N$_{y}$=N$_{z}$=150 or 170 \RC{N2 n'est pas dÃ©fini..}\RC{Nx est dÃ©fini, Ny? Nz?}
 \AG{La lettre 'x' n'est pas le symbole de la multiplication. Utiliser \texttt{\textbackslash times}.  Idem dans le texte, les figures, etc.}}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
 \label{tab:01}
 \end{center}
 \end{table}
 
+<<<<<<< HEAD
+In this section, we analyze the simulations conducted on various grid configurations presented in Table~\ref{tab:01}. 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...}
+=======
 
 
 
 
 In this  section, we analyze the  performance of algorithms running  on various
-grid configurations  (2x16, 4x8, 4x16  and 8x8). First,  the results in  Figure~\ref{fig:01}
+grid configurations  (2 $\times$ 16, 4 $\times$ 8, 4 $\times$ 16  and 8 $\times$ 8) and using an inter-network N2 defined in the test conditions in Table~\ref{tab:01}. 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...}
-
+%\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...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
 
 \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 input matrix size $N_{x}=150$ and $N_{x}=170$\RC{idem}
+  \caption{Various grid configurations with the matrix sizes 150$^3$ and 170$^3$
 \AG{Utiliser le point comme sÃ©parateur dÃ©cimal et non la virgule.  Idem dans les autres figures.}}
+\LZK{Pour quelle taille du problÃ¨me sont calculÃ©s les nombres d'itÃ©rations? Que reprÃ©sente le 2 Clusters x 16 Nodes with Nx=150 and Nx=170 en haut de la figure?}
   \label{fig:01}
 \end{figure}
 
-
+<<<<<<< HEAD
 The execution  times between  the two algorithms  is significant  with different
 grid architectures, even  with the same number of processors  (for example, 2x16
-and  4x8). We  can  observ  the low  sensitivity  of  the Krylov multisplitting  method
+and  4x8). We  can  observe  the low  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 2x16=32 to 8x8=64 processors. \RC{pas trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
+$40\%$ better (resp. $48\%$) when running from 2x16=32 to 8x8=64 processors. 
+\RC{pas trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
+\LZK{A revoir toute cette analyse... Le multi est plus performant que GMRES. Les temps d'exÃ©cution de multi sont sensibles au nombre de CLUSTERS. Il est moins performant pour un nombre grand de cluster. Avez vous d'autres remarques?}
+=======
+
+Secondly, 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  observ  the 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 reduction of the execution time for GMRES  (resp. Multisplitting)  algorithm is around $40\%$ (resp. around $48\%$) when running from 32 (grid 2 $\times$ 16) to 64 processors (grid 8 $\times$ 8) processors. \RC{pas trÃ¨s clair, c'est pas prÃ©cis de dire qu'un algo perform mieux qu'un autre, selon quel critÃ¨re?}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
 
-\subsubsection{Running on two different inter-clusters network speeds \\}
+\subsubsection{Simulations for two different inter-clusters network speeds \\}
 
 \begin{table} [ht!]
 \begin{center}
-\begin{tabular}{r c }
+\begin{tabular}{ll}
  \hline
- Grid Architecture & 2x16, 4x8\\ %\hline
- Network & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
+<<<<<<< HEAD
+ Grid architecture        & 2$\times$16, 4$\times$8\\ %\hline
+ \multirow{2}{*}{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}
+=======
+ Grid Architecture & 2 $\times$ 16, 4 $\times$ 8\\ %\hline
+ Inter Networks & N1 : bw=10Gbs-lat=8.10$^{-6}$ \\ %\hline
  - & N2 : bw=1Gbs-lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline
  \end{tabular}
-\caption{Test conditions: grid 2x16 and 4x8 with  networks N1 vs N2}
+\caption{Test conditions: grid 2 $\times$ 16 and 4 $\times$ 8 with  networks N1 vs N2}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
 \label{tab:02}
 \end{center}
 \end{table}
 
+<<<<<<< HEAD
 These experiments  compare the  behavior of  the algorithms  running first  on a
-speed inter-cluster  network (N1) and  also on  a less performant  network (N2). \RC{Il faut dÃ©finir cela avant...}
+slow inter-cluster  network (N1) and  also on  a more performant  network (N2). \RC{Il faut dÃ©finir cela avant...}
+=======
+In this section, the experiments  compare the  behavior of  the algorithms  running on a
+speeder inter-cluster  network (N1) and  also on  a less performant  network (N2) respectively defined in the test conditions Table~\ref{tab:02}. \RC{Il faut dÃ©finir cela avant...}
+>>>>>>> 2f78f080350308e2f46d8eff8d66a8e127fee583
 Figure~\ref{fig:02} shows that end users will reduce the execution time
-for  both  algorithms when using  a  grid  architecture  like  4x16 or  8x8: the reduction is about $2$. The results depict  also that when
+for  both  algorithms when using  a  grid  architecture  like  4 $\times$ 16 or  8 $\times$ 8: the reduction is about $2$. 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\%.
 
 
@@ -625,7 +671,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \begin{figure} [ht!]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
-\caption{Grid 2x16 and 4x8 with networks N1 vs N2
+\caption{Grid 2 $\times$ 16 and 4 $\times$ 8 with networks N1 vs N2
 \AG{\np{8E-6}, \np{5E-6} au lieu de 8E-6, 5E-6}}
 \label{fig:02}
 \end{figure}
@@ -638,7 +684,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
  Network & N1 : bw=1Gbs \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
@@ -674,7 +720,7 @@ magnitude with a latency of $8.10^{-6}$.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x16\\ %\hline
+ Grid Architecture & 2 $\times$ 16\\ %\hline
  Network & N1 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} \times N_{y} \times N_{z} =150 \times 150 \times 150$\\ \hline \\
  \end{tabular}
@@ -703,7 +749,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 4x8\\ %\hline
+ Grid Architecture & 4 $\times$ 8\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\
  Input matrix size & $N_{x}$ = From 40 to 200\\ \hline
  \end{tabular}
@@ -735,7 +781,7 @@ But the interesting results are:
 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 2x16 leading to the same conclusion.
+grid 2 $\times$ 16 leading to the same conclusion.
 
 \subsubsection{CPU Power impacts on performance}
 
@@ -743,7 +789,7 @@ grid 2x16 leading to the same conclusion.
 \centering
 \begin{tabular}{r c }
  \hline
- Grid architecture & 2x16\\ %\hline
+ Grid architecture & 2 $\times$ 16\\ %\hline
  Network & N2 : bw=1Gbs - lat=5.10$^{-5}$ \\ %\hline
  Input matrix size & $N_{x} = 150 \times 150 \times 150$\\ \hline
  \end{tabular}
@@ -804,7 +850,7 @@ The test conditions are summarized in the table~\ref{tab:07}: \\
 \centering
 \begin{tabular}{r c }
  \hline
- Grid Architecture & 2x50 totaling 100 processors\\ %\hline
+ Grid Architecture & 2 $\times$ 50 totaling 100 processors\\ %\hline
  Processors Power & 1 GFlops to 1.5 GFlops\\
    Intra-Network & bw=1.25 Gbits - lat=5.10$^{-5}$ \\ %\hline
    Inter-Network & bw=5 Mbits - lat=2.10$^{-2}$\\
