Richardson extrapolation allows truncation of higher order digital nets and sequences

TL;DR

This paper demonstrates that truncated higher order digital nets and sequences can replace polynomial lattice point sets in numerical integration, reducing computational precision requirements while maintaining near-optimal convergence rates.

Contribution

It introduces a novel approach using truncated digital nets for quadrature, enabling lower precision computations without sacrificing convergence speed.

Findings

Reduced digit requirements for quadrature points

Maintained almost optimal convergence rates

Applicable to high-dimensional, smooth function integration

Abstract

We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of convergence for numerical integration and can be constructed by the fast component-by-component search algorithm with smaller computational costs as compared to interlaced polynomial lattice rules. In this paper we prove that, instead of polynomial lattice point sets, truncated higher order digital nets and sequences can be used within the same algorithmic framework to explicitly construct good quadrature rules achieving the almost optimal rate of convergence. The major advantage of our new approach compared to original higher order digital nets is that we can significantly reduce the precision of points, i.e., the number of digits necessary to describe each quadrature node. This finding has a practically useful implication when either the number of points or the smoothness parameter is so large that original higher order digital nets require more than the available finite-precision floating point representations.