Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

TL;DR

This paper develops computable, non-asymptotic bounds for quasi-Monte Carlo integration using points with non-negative local discrepancy, extending classical results to higher dimensions and digital nets, with implications for high-dimensional numerical integration.

Contribution

It introduces methods to obtain computable upper and lower bounds for integrals using NNLD and NPLD points, generalizes Gabai's findings to digital nets in any dimension, and constructs high-dimensional NNLD lattice rules.

Findings

Derived non-asymptotic computable bounds for integrals.

Extended Gabai's NNLD property to digital nets in any dimension.

Constructed high-dimensional NNLD lattice rules with specific discrepancy properties.

Abstract

Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015) and let points $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^d$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^d$. An analogous non-positive local discrepancy (NPLD) property provides a computable lower bound. It has been known since Gabai (1967) that the two dimensional Hammersley points in any base $b\ge2$ have non-negative local discrepancy. Using the probabilistic notion of associated random variables, we generalize Gabai's finding to digital nets in any base $b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\ge3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\ge2$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.