Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights

Abstract

We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of r independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these r estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as r increases. Numerical experiments illustrate and support our theoretical findings.

Consistency of randomized integration methods

“…Remark 3.3. Let us also mention that recently median modified randomized quasi-Monte Carlo methods, see [PO21] and [GL22], attracted considerable attention.…”

Super-polynomial accuracy of one dimensional randomized nets using the median of means

“…Several uses in information based complexity are discussed in [15]. Uses in quasi-Monte Carlo include [26] metioned above as well as [13] for some laws of large numbers, [9] for some smoothness adaptive lattice rules and [8] for robust RQMC estimates.…”