Open type quasi-Monte Carlo integration based on Halton sequences in weighted Sobolev spaces

TL;DR

This paper develops open-type quasi-Monte Carlo methods using randomized Halton sequences in weighted Sobolev spaces, achieving optimal error bounds and strong polynomial tractability without complex search algorithms.

Contribution

It introduces an open-type QMC algorithm with randomized Halton sequences and p-adic shift, providing optimal error bounds and dimension-independent conditions.

Findings

Error bounds are optimal in the number of sample points.

Conditions are identified for dimension-independent error bounds.

Results are constructive, avoiding complex search algorithms.

Abstract

In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases $\boldsymbol{p}=(p_1,...,p_s)$ for which we study the root mean square (RMS) worst-case error. The randomization method is a $\boldsymbol{p}$-adic shift which is based on $\boldsymbol{p}$-adic arithmetic. The obtained error bounds are optimal in the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension $s$. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS $L_2$-discrepancy. Except for the $\boldsymbol{p}$-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.