The inverse of the star-discrepancy depends linearly on the dimension

Abstract

The inverse of the star-discrepancy depends linearly on the dimension

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

“…Although a bound of the form (1.14) indicates an ultimate order of convergence theoretically higher than the classical Monte Carlo rate of 1/ √ N, that bound is unsatisfactory when the dimensionality is high because, for fixed s, (ln N) s /N keeps growing with increasing N until N is exponentially large in s. In contrast to that somewhat negative observation is a remarkable result proved by Heinrich et al [12], which states that there exists a sequence of QMC point sets for which the star discrepancy is of order √ s/N with an unknown constant, or alternatively √ s ln s ln N/N with an explicit constant. However, no one knows yet how to construct QMC points that satisfy a bound of this kind.…”

Construction of Low-Discrepancy Point Sets of Small Size by Bracketing Covers and Dependent Randomized Rounding

“…Here the dependence of the inverse of the star discrepancy on d is optimal. This was also established in [HNWW01] by a lower bound for n * ∞ (d, ε), which was later improved by Hinrichs [Hin04] to n * ∞ (d, ε) ≥ c 0 dε −1 for 0 < ε < ε 0 , where c 0 , ε 0 > 0 are constants. The proof of (2) is not constructive but probabilistic, and the proof approach does not provide an estimate for the value of c. (A. Hinrichs presented a more direct approach to prove (2) with c = 10 at the Dagstuhl Seminar 04401 "Algorithms and Complexity for Continuous Problems" in 2004.…”

“…A bound more suitable for high-dimensional integration was established by Heinrich, Novak, Wasilkowski and Woźniakowski [HNWW01], who proved d * ∞ (n, d) ≤ cd 1/2 n −1/2 and n * ∞ (d, ε) ≤ c 2 dε −2 ,…”

Construction of low‐discrepancy point sets of small size

“…In the second bound the dependence on d is optimal [6,8]. A drawback here is that so far no reasonable estimate for the universal constant C has been published.…”

“…Therefore it is of interest to find useful upper bounds for the smallest possible star discrepancy of moderate sample sizes, and to construct small samples satisfying these bounds. In [6] Heinrich, Novak, Wasilkowski and Woźniakowski proved the bounds…”