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arXiv · 1602.00793

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

Josef Dick·Takashi Goda·Kosuke Suzuki·Takehito Yoshiki
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Abstract

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

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Josef Dick, Takashi Goda, Kosuke Suzuki, Takehito Yoshiki. 2016-02-02. Construction of interlaced polynomial lattice rules for infinitely differentiable functions. https://doi.org/10.1007/s00211-017-0882-x

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