The construction of good lattice rules and polynomial lattice rules

TL;DR

This paper reviews the construction of efficient lattice and polynomial lattice rules for high-dimensional integration, focusing on algebraic convergence rates and the impact of weights on error bounds.

Contribution

It provides a unified construction method for good lattice and polynomial lattice rules across various function spaces, including the special case of p=1.

Findings

Construction methods for good lattice rules are applicable for all p in (1, infinity]

Component-by-component algorithms enable efficient rule construction

Error bounds depend on the decay rate of the integrand's series representation

Abstract

A comprehensive overview of lattice rules and polynomial lattice rules is given for function spaces based on $\ell_p$ semi-norms. Good lattice rules and polynomial lattice rules are defined as those obtaining worst-case errors bounded by the optimal rate of convergence for the function space. The focus is on algebraic rates of convergence $O(N^{-\alpha+\epsilon})$ for $\alpha \ge 1$ and any $\epsilon > 0$, where $\alpha$ is the decay of a series representation of the integrand function. The dependence of the implied constant on the dimension can be controlled by weights which determine the influence of the different dimensions. Different types of weights are discussed. The construction of good lattice rules, and polynomial lattice rules, can be done using the same method for all $1 < p \le \infty$; but the case $p=1$ is special from the construction point of view. For $1 < p \le \infty$ the component-by-component construction and its fast algorithm for different weighted function spaces is then discussed.