Component-by-component construction of shifted Halton sequences

TL;DR

This paper develops a component-by-component algorithm to select effective p-adic shifts for Halton sequences, improving quasi-Monte Carlo integration accuracy in weighted Sobolev spaces.

Contribution

It introduces a finite set of candidate shifts for Halton sequences and provides an efficient algorithm to choose optimal shifts component-by-component.

Findings

Finite candidate shifts can achieve near-optimal error bounds.

Component-by-component algorithm effectively finds good shifts.

Improved error bounds in worst-case quasi-Monte Carlo integration.

Abstract

We study quasi-Monte Carlo integration in a weighted anchored Sobolev space. As the underlying integration nodes we consider Halton sequences in prime bases $\boldsymbol{p}=(p_1,\ldots,p_s)$ which are shifted with a $\boldsymbol{p}$-adic shift based on $\boldsymbol{p}$-adic arithmetic. The error is studied in the worst-case setting. In a recent paper, Hellekalek together with the authors of this article proved optimal error bounds in the root mean square sense, where the mean was extended over the uncountable set of all possible $\boldsymbol{p}$-adic shifts. Here we show that candidates for good shifts can in fact be chosen from a finite set and can be found by a component-by-component algorithm.