# Derandomised lattice rules for high dimensional integration

## TL;DR

This paper develops a deterministic shifted lattice rule approach for high-dimensional integration, showing existence of near-optimal shifts and demonstrating promising numerical results with a new CBC-based shift selection algorithm.

## Contribution

It introduces a deterministic shift method for lattice rules, proving the existence of near-optimal shifts and proposing a new CBC-based shift selection algorithm.

## Key findings

- Existence of a half-shifted rule with squared worst-case error close to the shift-averaged error.
- Numerical experiments show the effectiveness of the CBC for shift algorithm.
- Deterministic shifts perform comparably to random shifts in high-dimensional integration.

## 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. We show, by applying the principle that \emph{there is always at least one choice as good as the average}, 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 order only ${1/N^2}$. Numerical experiments, in which the generating vector is chosen component-by-component (CBC) as for randomly shifted lattices and then the shift by a new "CBC for shift" algorithm, yield encouraging results.