Digital Nets and Sequences Preprint

A complete preprint of the book

• Josef Dick and Friedrich Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010. © Cambridge University Press.

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).

Construction Algorithms for Higher Order Polynomial Lattice Rules

We recently submitted the manuscript

• J. Baldeaux, J. Dick, J. Greslehner, and F. Pillichshammer, Construction Algorithms for Higher Order Polynomial Lattice Rules. Submitted 2010.

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 . 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.

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 and be the adic expansion of (unique in the sense that infinitely many of the are different from ). We use the notation as in the book. Assume that is a prime number. For the sake of definiteness, consider the mean square worst-case error