An Intermediate Bound on the Star Discrepancy

Abstract

Let ${P}_{n}(z)$ denote the point set of an n-point rank-1 lattice rule with generating vector $z$. A criterion used to assess the ‘goodness’ of the point set ${P}_{n}(z)$ is the star discrepancy, ${D}^{{_\ast}}({P}_{n}(z))$. As calculating the star discrepancy is an NP-hard problem, then it is usual to work with bounds on it. In particular, it is known that the following two bounds hold:

$${D}^{{_\ast}}({P}_{ n}(z)) \leq 1 - {(1 - 1/n)}^{d} + T(z,n) \leq 1 - {(1 - 1/n)}^{d} + R(z,n)/2,$$

where d is the dimension and the quantities $T(z,n)$ and $R(z,n)$ are defined in the paper. Here we provide an intermediate bound on the star discrepancy by introducing a new quantity $W(z,n)$ which satisfies

$$T(z,n) \leq W(z,n) \leq R(z,n)/2.$$

Like $R(z,n)$, the quantity $W(z,n)$ may be calculated to a fixed precision in O(nd) operations. A component-by-component construction based on $W(z,n)$ is analysed. We present the results of numerical calculations which indicate that values of $W(z,n)$ are much closer to $T(z,n)$ than to $R(z,n)/2$.