Scrambled Polynomial Lattice Rules for Infinite-Dimensional Integration

TL;DR

This paper explores the use of scrambled polynomial lattice rules for infinite-dimensional integration, demonstrating improved tractability over previous methods based on scrambled digital nets.

Contribution

It introduces the application of scrambled polynomial lattice rules to infinite-dimensional integration and shows they outperform scrambled digital nets in strong tractability.

Findings

Enhanced strong tractability properties of scrambled polynomial lattice rules.

Improved error bounds for infinite-dimensional integration.

Demonstrated superiority over scrambled digital nets.

Abstract

In the random case setting, scrambled polynomial lattice rules as discussed in \cite{BD10} enjoy more favourable strong tractablility properties than scrambled digital nets. This short note discusses the application of scrambled polynomial lattice rules to infinite-dimensional integration. In \cite{HMGNR10}, infinite-dimensional integration in the random case setting was examined in detail, and results based on scrambled digital nets were presented. Exploiting these improved strong tractability properties of scrambled polynomial lattice rules and making use of the analysis presented in \cite{HMGNR10}, we improve on the results that were achieved using scrambled digital nets.