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


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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.

, 24 pages

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