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2007

Periodization strategy may fail in high dimensions

13 years 3 months ago
Periodization strategy may fail in high dimensions
Abstract. We discuss periodization of smooth functions f of d variables for approximation of multivariate integrals. The benefit of periodization is that we may use lattice rules, which have recently seen significant progress. In particular, we know how to construct effectively a generator of the rank-1 lattice rule with n points whose worst case error enjoys a nearly optimal bound Cd,p n−p . Here Cd,p is independent of d or depends at most polynomially on d, and p can be arbitrarily close to the smoothness of functions belonging to a weighted Sobolev space with an appropriate condition on the weights. If F denotes the periodization for f then the error of the lattice rule for a periodized function F is bounded by Cd,p n−p F with the norm of F given in the same Sobolev space. For small or moderate d, the norm of F is not much larger than the norm of f. This means that for small or moderate d, periodization is successful and allows us to use optimal properties of lattice rules al...
Frances Y. Kuo, Ian H. Sloan, Henryk Wozniakowski
Added 27 Dec 2010
Updated 27 Dec 2010
Type Journal
Year 2007
Where NA
Authors Frances Y. Kuo, Ian H. Sloan, Henryk Wozniakowski
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