The $b$-adic symmetrization of digital nets for quasi-Monte Carlo integration

TL;DR

This paper extends symmetrization techniques to digital nets over any base, analyzing their integration error and discrepancy, and demonstrates their effectiveness for high-dimensional quasi-Monte Carlo integration.

Contribution

It generalizes symmetrization from base 2 to arbitrary bases and studies its impact on QMC error and discrepancy in a unified framework.

Findings

Symmetrized digital nets achieve high-order convergence for smooth integrands.

Two-dimensional symmetrized Hammersley point sets attain optimal discrepancy order.

The approach enables construction of effective polynomial lattice rules.

Abstract

The notion of symmetrization, also known as Davenport's reflection principle, is well known in the area of the discrepancy theory and quasi-Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over $\mathbb{Z}_{b}$. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base $b\in \mathbb{N}$, $b\ge 2$. Subsequently, we study the QMC integration error of symmetrized digital nets over $\mathbb{Z}_{b}$ in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base $b$, and prove that the minimum Dick weight is large enough to achieve the best possible order of $L_p$ discrepancy for all $1\le p< \infty$.