Tag Archives: polynomial lattice rule

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).

The connection between the logarithmic Walsh degree of exactness, the t value of digital nets and an idea for future research

In this entry we show how the Walsh degree of a digital net is connected to its t value. This can lead to future research by using ideas developed for finding lattice rules with large trigonometric degree. Continue reading

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

Construction Algorithms for Higher Order Polynomial Lattice Rules

We recently submitted the manuscript

This paper fits into the work on higher order quasi-Monte Carlo rules which started with [22] and [31]. It can be viewed as the higher order extension of [7], where classical polynomial lattice rules were considered.

1. Construction Algorithms

As stated in the title of the manuscript, we present construction algorithms for higher order polynomial lattice rules. The construction is based on the worst-case error rather than the quality parameter {t}. This allows us to find good higher order polynomial lattice rules for weighted function spaces. It also presents a feasible alternative to the direct construction introduced in [31] (and [22] for the periodic case).

We use a variety of approaches to achieve our results. Continue reading

On the fast component-by-component algorithm for polynomial lattice rules

We present some simplification and observations concerning the fast component-by-component algorithm for polynomial lattice rules. This algorithm was introduced by Nuyens and Cools in

D. Nuyens and R. Cools. Fast component-by-component construction, a reprise for different kernels. In H. Niederreiter and D. Talay, editors, Monte Carlo and Quasi-Monte Carlo Methods 2004, pages 371–385. Springer-Verlag, 2006. Cited on pp. 82, 153, 178. See also Chapter 6 of Dirk’s PhD Thesis.

Let {\boldsymbol{x} = (x_1,\ldots, x_s)} and {x_i = \xi_{i,1} b^{-1} + \xi_{i,2} b^{-2} + \cdots} be the {b} adic expansion of {x} (unique in the sense that infinitely many of the {\xi_{i,j}} are different from {b-1}). We use the notation as in the book. Assume that {b} is a prime number. For the sake of definiteness, consider the mean square worst-case error Continue reading