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Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 163)

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.

Keywords

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

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Acknowledgments

The authors thank Art Owen for suggesting conducting the experimental comparisons presented here, his insightful discussions and his reviews of the manuscript.

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Correspondence to Colas Schretter .

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Schretter, C., He, Z., Gerber, M., Chopin, N., Niederreiter, H. (2016). Van der Corput and Golden Ratio Sequences Along the Hilbert Space-Filling Curve. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_28

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