Richardson extrapolation of polynomial lattice rules

TL;DR

This paper introduces extrapolated polynomial lattice rules, a new quasi-Monte Carlo method that achieves near-optimal convergence rates for multivariate integration of smooth functions, with efficient construction and practical advantages.

Contribution

The paper proposes a novel extrapolated polynomial lattice rule that improves construction cost and enables fast matrix-vector methods, advancing QMC integration techniques for smooth functions.

Findings

Achieves almost optimal convergence rate in Sobolev spaces.

Construction cost is reduced to order (s+α)N log N.

Enables fast QMC matrix-vector multiplication with theoretical guarantees.

Abstract

We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness $\alpha\geq 2$ defined over the $s$-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named \emph{extrapolated polynomial lattice rule}, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: i) construction of classical polynomial lattice rules over $\mathbb{F}_b$ with $\alpha$ consecutive sizes of nodes, $b^{m-\alpha+1},\ldots,b^{m}$, and ii) recursive application of Richardson extrapolation to a chain of $\alpha$ approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order $(s+\alpha)N\log N$, which improves the currently known result for interlaced polynomial lattice rule, that is of order $s\alpha N\log N$. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.