Worst-case error for unshifted lattice rules without randomisation
For researchers in numerical integration, this provides a theoretical foundation for unshifted lattice rules, removing the need for randomization, though the result is conditional on an unproven conjecture.
This paper proves the existence of unshifted lattice rules achieving worst-case error of order 1/√N (up to a dimension-independent logarithmic factor) for high-dimensional integration in an unanchored weighted Sobolev space, assuming a number-theoretic conjecture. Numerical evidence supports the conjecture.
An existence result is presented for the worst-case error of lattice rules for high dimensional integration over the unit cube, in an unanchored weighted space of functions with square-integrable mixed first derivatives. Existing studies rely on random shifting of the lattice to simplify the analysis, whereas in this paper neither shifting nor any other form of randomisation is considered. Given that a certain number-theoretic conjecture holds, it is shown that there exists an $N$-point rank-one lattice rule which gives a worst-case error of order $1/\sqrt{N}$ up to a (dimension-independent) logarithmic factor. Numerical results suggest that the conjecture is plausible.