Non-Uniform Low-Discrepancy Sequence Generation and Integration of Singular Integrands

Abstract

Summary.In this article, we will first highlight a method proposed by Hlawka and Mück to generate low-discrepancy sequences with an arbitrary distribution H and discuss its shortcomings. As an alternative, we propose an interpolated inversion method that is also shown to generate H-distributed low-discrepancy sequences, in an effort of order O(N log N ).Finally, we will address the issue of integrating functions with a singularity on the boundaries. Sobol' and Owen proved convergence theorems and orders for the uniform distribution, which we will extend to general distributions. Convergence orders will be proved under certain origin-or corner-avoidance conditions, as well as growth conditions on the integrand and the density. Our results prove that also non-uniform quasi-Monte Carlo methods can be well applied to integrands with a polynomial singularity at the integration boundaries.

Improved initialisation for centroidal Voronoi tessellation and optimal Delaunay triangulation

“…For CVT and ODT generation with uniform density a clear improvement in the resulting distribution could be demonstrated by using the Hammersley sequence. For non-uniform densities the effect is marginal and improved initialisations may be found by improving the inverse density computation in order to preserve the geometric properties of the Hammersley sequence better [34]. In future work, improvements for non-uniform tessellations and an extension to higher-dimensional domains may be achieved by better understanding the properties of the initialisation that help to improve the regularity of the results and the "landscape" of the energy functions for ODT and CVT.…”

Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

“…Chelson's generalized Koksma-Hlawka inequality is mentioned several times in the literature; it is explicitly stated, in differing forms, in [16,17,35,39,40,41,46,47]. In some of these instances it is stated without the transformation procedure and without using the incorrect identity (47) (which means that in those cases it is stated roughly in the same form as our Theorem 1 or Corollary 1).…”

“…, x N be a point set in [0,1] d . If µ is of product type, that is if it is the d-dimensional product measure of d one-dimensional measures, and if µ has a density (with respect to λ), then in [16,21] transformation methods are presented which (in a computationally tractable way) generate a sequence y 1 , . .…”

A Pragmatic View on Numerical Integration of Unbounded Functions

“…In [HK05] Hartinger and Kainhofer deal with the problem of generating low discrepancy sequences with an arbitrary distribution H. While they did so before ([HKT04]) they identify some disadvantages which carry over to the transformed sequence they proposed. They specifically deal with the property of the Hlawka-Mück method that for some applications the points of the generated sequence of a set with cardinality N lie on a lattice with spacing 1/N .…”