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The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets

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

Abstract

In this paper, we study randomized quasi-Monte Carlo (QMC) integration using digitally shifted digital nets. We express the mean square QMC error of the nth discrete approximation \(f_n\) of a function \(f:[0,1)^s\rightarrow \mathbb {R}\) for digitally shifted digital nets in terms of the Walsh coefficients of f. We then apply a bound on the Walsh coefficients for sufficiently smooth integrands to obtain a quality measure called Walsh figure of merit for the root mean square error, which satisfies a Koksma–Hlawka type inequality on the root mean square error. Through two types of experiments, we confirm that our quality measure is of use for finding digital nets which show good convergence behavior of the root mean square error for smooth integrands.

Keywords

  • Randomized quasi-Monte Carlo
  • Digital shift
  • Digital net
  • Walsh functions
  • Walsh figure of merit

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Acknowledgments

The authors would like to thank Prof. Makoto Matsumoto for helpful discussions and comments. The work of T.G. was supported by Grant-in-Aid for JSPS Fellows No.24-4020. The works of R.O., K.S. and T.Y. were supported by the Program for Leading Graduate Schools, MEXT, Japan. The work of K.S. was partially supported by Grant-in-Aid for JSPS Fellows Grant number 15J05380.

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Correspondence to Kosuke Suzuki .

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Goda, T., Ohori, R., Suzuki, K., Yoshiki, T. (2016). The Mean Square Quasi-Monte Carlo Error for Digitally Shifted Digital Nets. 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_16

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