Tractability of Multivariate Integration for Weighted Korobov Classes

We study the worst-case error of multivariate integration in weighted Korobov classes of periodic functions of d coordinates. This class is defined in terms of weights γ_j which moderate the behavior of functions with respect to successive coordinates. We study two classes of quadrature rules. They are quasi-Monte Carlo rules which use n function values and in which all quadrature weights are 1/n and rules for which all quadrature weights are non-negative. Tractability for these two classes of quadrature rules means that the minimal number of function values needed to guarantee error in the worst-case setting is bounded by a polynomial in d and ⁻¹. Strong tractability means that the bound does not depend on d and depends polynomially on ⁻¹. We prove that strong tractability holds iff ∑^∞_j=1 γ_j<∞, and tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/log d<∞. Furthermore, strong tractability or tractability results are achieved by the relatively small class of lattice rules. We also prove that if ∑^∞_j=1 γ^1/α_j<∞, where α measures the decay of Fourier coefficients in the weighted Korobov class, then for d1, n prime and δ>0 there exist lattice rules that satisfy an error bound independent of d and of order n^−α/2+δ. This is almost the best possible result, since the order n^−α/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ∑^∞_j=1 γ^1/2_j<∞, then for d1, n prime and δ>0 there exist shifted lattice rules that satisfy an error bound independent of d and of order n^−1+δ. We also check how the randomized error of the (classical) Monte Carlo algorithm depends on d for weighted Korobov classes. It turns out that Monte Carlo is strongly tractable iff ∑^∞_j=1 log γ_j<∞ and tractable iff lim sup_d→∞ ∑^d_j=1 log γ_j/log d<∞. Hence, in particular, for γ_j=1 we have the usual Korobov space in which integration is intractable for the two classes of quadrature rules in the worst-case setting, whereas Monte Carlo is strongly tractable in the randomized setting.