Quasi-Monte Carlo with a Hankel random digital net

TL;DR

This paper introduces a novel randomized digital net design using Hankel matrices, simplifying construction and maintaining good convergence, with theoretical analysis and numerical validation.

Contribution

It presents a new Hankel matrix-based randomized digital net design that reduces complexity while preserving convergence properties.

Findings

The proposed design simplifies digital net construction.

The method achieves desirable convergence rates.

Numerical experiments confirm theoretical results.

Abstract

This paper proposes a new randomized design of digital nets in which the generating matrices are chosen to be random Hankel matrices. Compared with previous randomized designs of digital nets, this approach simplifies the construction process and reduces the number of random variables required, while still achieving desirable convergence rates when combined with appropriate estimators. We analyze the properties of the proposed design, derive bounds for Walsh coefficients, and provide error analysis for both the median-of-means estimator and a newly proposed greedy selection estimator, i.e. the selection of the best design from a batch in terms of a worst-case error bound. Numerical experiments validate our theoretical findings and demonstrate the practical performance of the proposed methods.