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@@ -2,7 +2,8 @@
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\documentclass{llncs}
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\input{prelude}
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\begin{document}
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-\title{$\mbox{\EightStarBold}$ Discrepancies for generalized Halton points}
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+\title{$\mbox{\EightStarBold}$ Discrepancies for generalized Halton points\\
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+Comparison of three heuristics for generating points set}
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% \titlerunning{} % abbreviated title (for running head)
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% also used for the TOC unless
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% \toctitle is used
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@@ -19,11 +20,25 @@
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\maketitle
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\makeatletter
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-
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+\renewcommand\bibsection%
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+{
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+ \section*{\refname
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+ \@mkboth{\MakeUppercase{\refname}}{\MakeUppercase{\refname}}}
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+ }
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\makeatother
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\begin{abstract}
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+ Geometric discrepancies are standard measures to quantify the irregularity of
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+ distributions. They are an important notion in numerical integration.
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+ One of the most important discrepancy notions is the so-called star
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+ discrepancy. Roughly speaking, a point set of low star discrepancy value
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+ allows for a small approximation error in quasi-Monte Carlo integration.
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+ In this work we present a tool realizing the implantation of three
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+ basics heuristics for construction low discrepancy points sets
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+ in the generalized Halton model: fully random search, local search with
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+ simmulated annealing and genetic $(5+5)$ search with a ad-hoc
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+ crossover function.
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\end{abstract}
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@@ -351,9 +366,9 @@ annealing and genetic search --- is clear over fully random search.
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Both curves for these heuristics are way below the error band of random
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search. As a result \emph{worse average results of non trivial heuristics are
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better than best average results when sampling points at random}.
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-In dimension 2~\ref{wrap2z}, the best results are given by the genetic search,
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+In dimension 2~\ref{wrap2z}, the best results are given by the simulated annealing,
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whereas in dimension 3 and 4~\ref{wrap3z},~\ref{wrap4z}, best results are
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-given by simulated annealing. It is also noticeable that in that range
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+given by genetic search. It is also noticeable that in that range
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of points the error rates are roughly the same for all heuristics:
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\emph{for 1000 iteration, the stability of the results is globally the
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same for each heuristic}.
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@@ -384,6 +399,13 @@ same for each heuristic}.
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\section{Conclusion}
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+\section*{Acknoledgments}
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+We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
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+for providing an implementation of the DEM algorithm [DEM96].
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+We would also like to thank Christoff Durr and Carola Doerr
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+for several very helpful talks on the topic of this work.
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+Both Thomas Espitau and Olivier Marty supported by the French Ministry for
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+Research and Higher Education, trough the Ecole Normale Supérieure.
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\bibliographystyle{alpha}
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\bibliography{bi}
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\end{document}
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