Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

TL;DR

This paper introduces a fast algorithm for constructing higher order polynomial lattice rules, significantly improving the efficiency of worst-case error computation for high-dimensional quadrature of smooth functions.

Contribution

It presents a novel, efficient component-by-component construction method exploiting cyclic group structure, enhancing the performance of higher order polynomial lattice rules.

Findings

The new rules outperform existing higher order digital net-based quadrature rules.

Explicit formulas for base 2 are provided for practical implementation.

Numerical results demonstrate improved worst-case error performance.

Abstract

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets.