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

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.