Quasi-Monte Carlo integration using digital nets with antithetics

TL;DR

This paper extends antithetic sampling to digital nets over any base for quasi-Monte Carlo integration, demonstrating improved convergence rates for smooth functions and providing theoretical and numerical insights into variance reduction techniques.

Contribution

It generalizes antithetic sampling from base 2 to arbitrary bases in digital nets and analyzes its impact on QMC error, including the existence of good point sets with antithetics.

Findings

Antithetic sampling improves convergence for smooth integrands.

Theoretical proof of existence of good higher order polynomial lattice point sets with antithetics.

Numerical experiments show enhanced convergence rates up to 100 dimensions.

Abstract

Antithetic sampling, which goes back to the classical work by Hammersley and Morton (1956), is one of the well-known variance reduction techniques for Monte Carlo integration. In this paper we investigate its application to digital nets over $\mathbb{Z}_b$ for quasi-Monte Carlo (QMC) integration, a deterministic counterpart of Monte Carlo, of functions defined over the $s$-dimensional unit cube. By looking at antithetic sampling as a geometric technique in a compact totally disconnected abelian group, we first generalize the notion of antithetic sampling from base $2$ to an arbitrary base $b\ge 2$. Then we analyze the QMC integration error of digital nets over $\mathbb{Z}_b$ with $b$-adic antithetics. Moreover, for a prime $b$, we prove the existence of good higher order polynomial lattice point sets with $b$-adic antithetics for QMC integration of smooth functions in weighted Sobolev spaces. Numerical experiments based on Sobol' point sets up to $s=100$ show that the rate of convergence can be improved for smooth integrands by using antithetic sampling technique, which is quite encouraging beyond the reach of our theoretical result and motivates future work to address.