# Tag Archives: numerical integration

## 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).

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
Recently I submitted the paper titled Quasi-Monte Carlo numerical integration over $\mathbb{R}^s$: digital nets and worst-case error. I will give some heuristic explanation of the results in this paper. Interesting in this context is also the sparse grid approach from the PhD thesis of Markus Holtz (supervised by Prof. Michael Griebel).
Therein I aim to develop a theory on quasi-Monte Carlo integration over $\mathbb{R}^s$. The idea is to transform a digital net from $[0,1]^s$ to $\mathbb{R}^s$ such that in elementary intervals of the form $\displaystyle \prod_{i=1}^s [A b^j, (A+1) b^j)$ one has a digitally shifted digital net, and at the same time, globally one has a given (discretized) distribution. Continue reading