Van der Corput and Golden Ratio sequences along the Hilbert space filling curve

Colas Schretter, Zhijian He, Mathieu Gerber, Nicolas Chopin, Harald Niederreiter

Research output: Chapter in Book/Report/Conference proceedingChapter in a book

Abstract

This work investigates the star discrepancies and squared integration errors of two quasi-random points constructions using a generator one-dimensional sequence and the Hilbert space-filling curve. This recursive fractal is proven to maximize locality and passes uniquely through all points of the d-dimensional space. The van der Corput and the golden ratio generator sequences are compared for randomized integro-approximations of both Lipschitz continuous and piecewise constant functions. We found that the star discrepancy of the construction using the van der Corput sequence reaches the theoretical optimal rate when the number of samples is a power of two while using the golden ratio sequence performs optimally for Fibonacci numbers. Since the Fibonacci sequence increases at a slower rate than the exponential in base 2, the golden ratio sequence is preferable when the budget of samples is not known beforehand. Numerical experiments confirm this observation.
Original languageEnglish
Title of host publicationMonte Carlo and Quasi-Monte Carlo Methods
Subtitle of host publicationMCQMC, Leuven, Belgium, April 2014
PublisherSpringer International Publishing AG
ISBN (Electronic)978-3-319-33507-0
ISBN (Print)978-3-319-33505-6
Publication statusPublished - 2016

Publication series

NameSpringer Proceedings in Mathematics & Statistics

Keywords

  • Quasi-random points
  • Hilbert curve
  • discrepancy
  • golden ratio sequence
  • numerical integration

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