<abstract xmlns="http://www.w3.org/1999/xhtml"> 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 ℤb. 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 ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤb 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 Lp discrepancy for all 1 ≤ p &lt; ∞.</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 ℤb. 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 ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤb 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 Lp discrepancy for all 1 ≤ p < ∞.

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration"}

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)...

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 1 - 25

Received: Nov 04, 2015

Accepted: Nov 30, 2015

DOI: https://doi.org/10.1515/udt-2017-0001

Keywords
Quasi-Monte Carlo, b-adic symmetrization, digital nets, Hammersley point sets

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Article Category: Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Published Online: Jul 22, 2017

Page range: 1 - 25

Received: Nov 04, 2015

Accepted: Nov 30, 2015

DOI: https://doi.org/10.1515/udt-2017-0001

KeywordsQuasi-Monte Carlo, b-adic symmetrization, digital nets, Hammersley point sets

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Quasi-Monte Carlo, b-adic symmetrization, digital nets, Hammersley point sets