Construction of interlaced polynomial lattice rules for infinitely differentiable functions

TL;DR

This paper constructs interlaced polynomial lattice rules for high-dimensional integration of infinitely differentiable functions, achieving super-polynomial convergence independent of dimension through a fast algorithm.

Contribution

It provides a constructive method to build QMC rules with super-polynomial convergence for infinitely differentiable functions, using interlaced polynomial lattice rules and a component-by-component algorithm.

Findings

Achieves dimension-independent super-polynomial convergence

Develops a fast construction algorithm with $O(sN(\log N)^2)$ complexity

Uses a novel Jensen's inequality variant in error analysis

Abstract

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.