Tractability properties of the weighted star discrepancy of the Halton sequence

TL;DR

This paper demonstrates that the Halton sequence and similar digital sequences achieve strong polynomial tractability for weighted star discrepancy under a mild weight condition, advancing understanding of their efficiency in high-dimensional integration.

Contribution

It establishes the first explicit condition under which the Halton sequence attains strong polynomial tractability for weighted star discrepancy, extending to Niederreiter and other digital sequences.

Findings

Halton sequence achieves strong polynomial tractability under mild weight conditions.

Results extend to Niederreiter and other digital sequences.

Applicable to weighted unanchored discrepancy.

Abstract

We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(\gamma_j)_{j \ge 1}$ under the mildest condition on the weight sequence known so far for explicitly constructive sequences. The condition requires $\sup_{d \ge 1} \max_{\emptyset \not= \mathfrak{u} \subseteq [d]} \prod_{j \in \mathfrak{u}} (j \gamma_j) < \infty$. The same result holds for Niederreiter sequences and for other types of digital sequences. Our results are true also for the weighted unanchored discrepancy.