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...

• Type: Journal
• Year: 2007
• Where: NA
• Authors: Frances Y. Kuo, Ian H. Sloan, Henryk Wozniakowski
• Keywords: Lattice Rules, NA 2007, Periodization, Rank-1 Lattice Rule