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 C _d,p n ^−p. Here C _d,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 C _d,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 also for non-periodic functions. The situation is quite different if d is large since the norm of F can be exponentially larger than the norm of f. This can already be seen for f = 1. Hence, the upper bound of the worst case error of the lattice rule for periodized functions is quite bad for large d. We conjecture not only that this upper bound is bad, but also that all lattice rules fail for large d. That is, if we fix the number of points n and let d go to infinity then the worst case error of any lattice rule is bounded from below by a positive constant independent of n. We present a number of cases suggesting that this conjecture is indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.