\usepackage{amsmath}
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+\usepackage{graphicx}
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The present paper is organized as follows. In Section~\ref{sec:02} some related
works are presented. Section~\ref{sec:03} presents our two-stage algorithm using
a least-square residual minimization. Section~\ref{sec:04} describes some
-convergence results on this method. In Section~\ref{sec:05}, parallization
-details of TSARM are given. Section~\ref{sec:06} shows some experimental
+convergence results on this method. Section~\ref{sec:05} shows some experimental
results obtained on large clusters of our algorithm using routines of PETSc
toolkit. Finally Section~\ref{sec:06} concludes and gives some perspectives.
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\item $\epsilon_{ls}$ the threshold to stop the least-square method
\end{itemize}
-%%%*********************************************************
-%%%*********************************************************
-
-\section{Convergence results}
-\label{sec:04}
-
-
-
-%%%*********************************************************
-%%%*********************************************************
-\section{Parallelization}
-\label{sec:05}
The parallelisation of TSARM relies on the parallelization of all its
parts. More precisely, except the least-square step, all the other parts are
classical operations: dots, norm, multiplication and addition on vectors. All
these operations are easy to implement in PETSc or similar environment.
+
+
+%%%*********************************************************
+%%%*********************************************************
+
+\section{Convergence results}
+\label{sec:04}
+
+
+
+
%%%*********************************************************
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\section{Experiments using petsc}
-\label{sec:06}
+\label{sec:05}
In order to see the influence of our algorithm with only one processor, we first
-Larger experiments ....\\
In the following we describe the applications of PETSc we have experimented. Those applications are available in the ksp part which is suited for scalable linear equations solvers:
\begin{itemize}
-\item ex15 is an example which solves in parallel a 2D homogeneous
- Laplacian. Thediagonal is equals to 4 and 4 extra-diagonals representing the
- neighbors in each directions is equal to -1. This example is used in many
- physical phenomena , for exemple, heat and fluid flow, wave propagation...
-\item
+\item ex15 is an example which solves in parallel an operator using a finite difference scheme. The diagonal is equals to 4 and 4
+ extra-diagonals representing the neighbors in each directions is equal to
+ -1. This example is used in many physical phenomena , for exemple, heat and
+ fluid flow, wave propagation...
+\item ex54 is another example based on 2D problem discretized with quadrilateral finite elements. For this example, the user can define the scaling of material coefficient in embedded circle, it is called $\alpha$.
\end{itemize}
-
+For more technical details on these applications, interested reader are invited
+to read the codes available in the PETSc sources. Those problem have been
+chosen because they are scalable with many cores. We have tested other problem
+but they are not scalable with many cores.
\end{table*}
+\begin{figure}
+\centering
+ \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03}}
+\label{fig:01}
+\end{figure}
+
+
+
+
+
\begin{table*}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
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\section{Conclusion}
-\label{sec:07}
+\label{sec:06}
%The conclusion goes here. this is more of the conclusion
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