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Walsh Figure of Merit for Digital Nets: An Easy Measure for Higher Order Convergent QMC

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

Abstract

Fix an integer s. Let \(f:[0,1)^s \rightarrow {\mathbb {R}}\) be an integrable function. Let \({\mathscr {P}}\subset [0, 1]^s\) be a finite point set. Quasi-Monte Carlo integration of f by \({\mathscr {P}}\) is the average value of f over \({\mathscr {P}}\) that approximates the integration of f over the s-dimensional cube. Koksma–Hlawka inequality tells that, by a smart choice of \({\mathscr {P}}\), one may expect that the error decreases roughly \(O(N^{-1}(\log N)^s)\). For any \(\alpha \ge 1\), J. Dick gave a construction of point sets such that for \(\alpha \)-smooth f, convergence rate \(O(N^{-\alpha }(\log N)^{s\alpha })\) is assured. As a coarse version of his theory, M-Saito-Matoba introduced Walsh figure of Merit (WAFOM), which gives the convergence rate \(O(N^{-C\log N/s})\). WAFOM is efficiently computable. By a brute-force search of low WAFOM point sets, we observe a convergence rate of order \(N^{-\alpha }\) with \(\alpha >1\), for several test integrands for \(s=4\) and 8.

Keywords

  • Quasi-Monte Carlo
  • Walsh figure of merit
  • Numerical integration
  • Digital nets

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Notes

  1. 1.

    See Sect. 2.3 for a definition of digital nets; there we use the italic P instead of \({\mathscr {P}}\) for a digital net, to stress that actually P is a subspace of a discrete space, while \({\mathscr {P}}\) is in a continuous space \(I^s\).

  2. 2.

    If f has Lipschitz constant C, namely, satisfies \(f(x-y)<C|x-y|\), then the error is bounded by \(C\sqrt{s}2^{-n}\) [16, Lemma 2.1].

  3. 3.

    The perpendicular space is called “the dual space” in most literatures on QMC and coding theory. However, in pure algebra, the dual space to a vector space V over a field k means \(V^*:=\mathrm {Hom}_k(V,k)\), which is defined without using inner product. In this paper, we use the term “perpendicular” going against the tradition in this area.

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Acknowledgments

The authors are deeply indebted to Josef Dick, who patiently and generously informed us of beautiful researches in this area, and to Harald Niederreiter for leading us to this research. They thank for the indispensable helps by the members of Komaba-Applied-Algebra Seminar (KAPALS): Takashi Goda, Shin Harase, Shinsuke Mori, Syoiti Ninomiya, Mutsuo Saito, Kosuke Suzuki, and Takehito Yoshiki. We are thankful to the referees, who informed of numerous improvements on the manuscript. The first author is partially supported by JST CREST, JSPS/MEXT Grant-in-Aid for Scientific Research No.21654017, No.23244002, No.24654019, and No.15K13460. The second author is partially supported by the Program for Leading Graduate Schools, MEXT, Japan.

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Correspondence to Makoto Matsumoto .

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Matsumoto, M., Ohori, R. (2016). Walsh Figure of Merit for Digital Nets: An Easy Measure for Higher Order Convergent QMC. 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_5

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