Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

TL;DR

This paper proves that lattice and polynomial lattice rules constructed by the CBC algorithm maintain near-optimal convergence rates and tractability across various smoothness and weight classes, extending their applicability in high-dimensional QMC integration.

Contribution

The paper establishes the stability and near-optimal convergence of CBC-constructed lattice and polynomial lattice rules for broader function classes with different smoothness and weights.

Findings

Lattice rules achieve almost optimal convergence for general smoothness and weights.

Polynomial lattice rules exhibit similar stability in weighted Walsh spaces.

Bounds on weighted star discrepancy are provided, supporting tractability analysis.

Abstract

We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights. In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.