Derandomised lattice rules for high dimensional integration
This work provides a derandomized alternative to randomly shifted lattice rules, offering theoretical guarantees for deterministic shifts in high-dimensional numerical integration.
The paper introduces a deterministic 'half-shifted' lattice rule for high-dimensional integration, showing that for a given generating vector, the squared worst-case error differs from the shift-averaged error by only O(1/N^2). Numerical experiments with a new CBC-for-shift algorithm yield encouraging results.
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.