Where are the logs?

TL;DR

This paper investigates the error rates of Quasi-Monte Carlo (QMC) integration for fixed integrands, showing that improved rates are possible under certain conditions, and provides empirical evidence supporting these theoretical findings.

Contribution

It demonstrates that for fixed integrands, QMC error rates can be better than previously thought, using advanced theoretical techniques and empirical testing.

Findings

Error rates with $r<(d-1)/2$ are achievable for fixed integrands.

Empirical tests in 2D show no need for $r>1$.

Potential need for $r>1$ in 3D with large $n$.

Abstract

The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\log(n)^r)$ with $r=d$ for extensible sequences and $r=d-1$ otherwise. Such rates hold uniformly over all $d$ dimensional integrands of Hardy-Krause variation one when using $n$ evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on $n$. In this paper we show that rates with any $r<(d-1)/2$ can hold when $f$ is held fixed as $n\to\infty$. This is accomplished following a suggestion of Erich Novak to use some unpublished results of Trojan from the 1980s as given in the information based complexity monograph of Traub, Wasilkowski and Wo\'zniakowski. The proof is made by applying a technique of Roth with the theorem of Trojan. The proof is non constructive and we do not know of any integrand of bounded variation in the sense of Hardy and Krause for which the QMC error exceeds $(\log n)^{1+\epsilon}/n$ for infinitely many $n$ when using a digital sequence such as one of Sobol's. An empirical search when $d=2$ for integrands designed to exploit known weaknesses in certain point sets showed no evidence that $r>1$ is needed. An example with $d=3$ and $n$ up to $2^{100}$ might possibly require $r>1$.