
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 with for extensible sequences and otherwise. Such rates hold uniformly over all dimensional integrands of Hardy-Krause variation one when using evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on . In this paper we show that rates with any can hold when is held fixed as . This is accomplished following a suggestion of Erich Novak to use some unpublished results of Trojan from the 1980s as given in the information based complexity monograph of Traub, Wasilkowski and Wo\'zniakowski.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…
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Taxonomy
TopicsMathematical Approximation and Integration · Cryptography and Residue Arithmetic · Coding theory and cryptography
