# Derandomised lattice rules for high dimensional integration

## DOI:

https://doi.org/10.21914/anziamj.v60i0.14110## Keywords:

lattice rules, quasiâ€“Monte Carlo methods## Abstract

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of \(1/(2N)\) where \(N\) is the number of points in the lattice. By applying the principle that*there is always at least one choice as good as the average*, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order \({1/N^2}\). We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging.

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## Published

2019-11-16

## Issue

## Section

Proceedings Computational Techniques and Applications Conference