X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rce2015.git/blobdiff_plain/976b793822811b4335b31747dc0e5542a41fc62b..96e726f9524ffdb8eaead14a943d80870ea1da65:/paper.tex

diff --git a/paper.tex b/paper.tex
index 0e69066..c3cfdbb 100644
--- a/paper.tex
+++ b/paper.tex
@@ -321,7 +321,7 @@ A_{\ell\ell} x_\ell = c_\ell,\mbox{~for~}\ell=1,\ldots,L,
 \end{equation}
 where right-hand sides $c_\ell=b_\ell-\sum_{m\neq\ell}A_{\ell m}x_m$ are computed using the shared vectors $x_m$. In this paper, we use the well-known iterative method GMRES~\cite{saad86} as an inner iteration to approximate the solutions of the different splittings arising from the block Jacobi multisplitting of matrix $A$. The algorithm in Figure~\ref{alg:01} shows the main key points of our block Jacobi two-stage method executed by a cluster of processors. In line~\ref{solve}, the linear sub-system~(\ref{eq:03}) is solved in parallel using GMRES method where $\MIG$ and $\TOLG$ are the maximum number of inner iterations and the tolerance threshold for GMRES respectively. The convergence of the two-stage multisplitting methods, based on synchronous or asynchronous iterations, has been studied by many authors for example~\cite{Bru95,bahi07}.
 
-\begin{figure}[t]
+\begin{figure}[htpb]
 %\begin{algorithm}[t]
 %\caption{Block Jacobi two-stage multisplitting method}
 \begin{algorithmic}[1]
@@ -359,7 +359,7 @@ At each $s$ outer iterations, the algorithm computes a new approximation $\tilde
 \end{equation}
 The algorithm in Figure~\ref{alg:02} includes the procedure of the residual minimization and the outer iteration is restarted with a new approximation $\tilde{x}$ at every $s$ iterations. The least-squares problem~(\ref{eq:06}) is solved in parallel by all clusters using CGLS method~\cite{Hestenes52} such that $\MIC$ is the maximum number of iterations and $\TOLC$ is the tolerance threshold for this method (line~\ref{cgls} in Figure~\ref{alg:02}).
 
-\begin{figure}[t]
+\begin{figure}[htbp]
 %\begin{algorithm}[t]
 %\caption{Krylov two-stage method using block Jacobi multisplitting}
 \begin{algorithmic}[1]
@@ -591,7 +591,7 @@ the Krylov two-stage algorithm.
 %\RC{Les légendes ne sont pas explicites...}
 %\RCE{Corrige}
 
-\begin{figure} [ht!]
+\begin{figure} [htbp]
   \begin{center}
     \includegraphics[width=100mm]{cluster_x_nodes_nx_150_and_nx_170.pdf}
   \end{center}
@@ -641,7 +641,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 
 
 %\begin{wrapfigure}{l}{100mm}
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{cluster_x_nodes_n1_x_n2.pdf}
 \caption{Various grid configurations with networks N1 vs N2
@@ -667,7 +667,7 @@ the  network speed  drops down (variation of 12.5\%), the  difference between  t
 \label{tab:03}
 \end{table}
 
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{network_latency_impact_on_execution_time.pdf}
 \caption{Network latency impacts on execution time
@@ -700,12 +700,12 @@ more  than $75\%$  (resp.  $82\%$)  of the  execution  for  the classical  GMRES
 \end{table}
 
 
-\begin{figure} [ht!]
+\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}
+\caption{Network bandwith impacts on execution time}
+%\AG{``Execution time'' avec un 't' minuscule}. Idem autres figures.}
+%\RCE{Corrige}
 \label{fig:04}
 \end{figure}
 
@@ -730,7 +730,7 @@ of $40\%$ which is only around $24\%$ for the classical GMRES.
 \end{table}
 
 
-\begin{figure} [ht!]
+\begin{figure} [htbp]
 \centering
 \includegraphics[width=100mm]{pb_size_impact_on_execution_time.pdf}
 \caption{Problem size impacts on execution time}
@@ -759,7 +759,7 @@ grid 2 $\times$ 16 leading to the same conclusion.
 
 \subsubsection{CPU Power impacts on performance}
 
-\begin{table} [ht!]
+\begin{table} [htbp]
 \centering
 \begin{tabular}{r c }
  \hline
@@ -811,18 +811,19 @@ synchronization  with   the  other   processors.  Thus,  the   asynchronous  may
 theoretically reduce  the overall execution  time and can improve  the algorithm
 performance.
 
-\RC{la phrase suivante est bizarre, je ne comprends pas pourquoi elle vient ici}
-\RCE{C est la description du dernier test sync/async avec l'introduction de la notion de relative gain}
-In this section, Simgrid simulator tool has been successfully used to show
-the efficiency of  the multisplitting in asynchronous mode and  to find the best
-combination of the grid resources (CPU,  Network, input matrix size, \ldots ) to
-get    the   highest    \textit{"relative    gain"}   (exec\_time$_{GMRES}$    /
-exec\_time$_{multisplitting}$) in comparison with the classical GMRES time.
+In this section,  the Simgrid simulator is  used to compare the  behavior of the
+multisplitting in  asynchronous mode  with GMRES  in synchronous  mode.  Several
+benchmarks have  been performed with  various combination of the  grid resources
+(CPU, Network, input  matrix size, \ldots ). The test  conditions are summarized
+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 version is faster than GMRES.
 
 
-The test conditions are summarized in the table~\ref{tab:07}: \\
 
-\begin{table} [ht!]
+\begin{table} [htbp]
 \centering
 \begin{tabular}{r c }
  \hline
@@ -872,7 +873,7 @@ geographically distant clusters through the internet.
     power (GFlops)
     & 1    & 1    & 1    & 1.5       & 1.5  & 1.5         & 1.5         & 1         & 1.5       & 1.5 \\
     \hline
-    size (N)
+    size ($N^3$)
     & 62  & 62   & 62        & 100       & 100 & 110       & 120       & 130       & 140       & 150 \\
     \hline
     Precision