Tag Archives: randomised quasi-Monte Carlo

Digital Nets and Sequences Preprint

A complete preprint of the book

is now available here. The final published version can be obtained directly from Cambridge University Press here.

The preprint version differs of course from the final version, for instance, the page numbers are different. However, the numbering of Chapters, Sections, Theorems, Lemmas, Corollaries, Definitions and Examples is the same in both versions. The list of corrections is for the published version. We do not have a separate list for the preprint version (though the corrections for the published version also apply to the preprint version).

A Construction of Polynomial Lattice Rules with Small Gain Coefficients

Recently J. Baldeaux and myself submitted the manuscript titled A Construction of Polynomial Lattice Rules with Small Gain Coefficients.

In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Art Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of N^{-(2\alpha + 1) +\delta}, for all \delta >0, assuming that the function under consideration bounded fractional variation of order \alpha and where N denotes the number of quadrature points. We give some further comments on the paper. Continue reading

Consistency of Markov Chain quasi-Monte Carlo for continuous state spaces

This post is based on the paper

and my previous talks on this topic at the UNSW statistic seminar and the Dagstuhl Workshop in 2009. The slides of my talk at Dagstuhl can be found here. I give an illustration of the results rather than rigorous proofs, which can be found in the paper [CDO].

The classical paper on this topic is by [Chentsov 1967]:

N. N. Chentsov, Pseudorandom numbers for modelling Markov chains, Computational Mathematics and Mathematical Physics, 7, 218–2332, 1967.

Further important steps were taken by A. Owen and S. Tribble, see doi: 10.1073/pnas.0409596102 and doi: 10.1214/07-EJS162. Recent papers of interest in this context are also by A. Hinrichs doi:10.1016/j.jco.2009.11.003, and D. Rudolf doi: 10.1016/j.jco.2008.05.005. The slides of the presentations at Dagstuhl of Hinrichs can be found here here and of Rudolf can be found here.

1. Introduction

The task is to approximate an integral Continue reading