Tractability of multivariate integration for weighted Korobov classes

Abstract

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 γj 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 e in the worst-case setting is bounded by a polynomial in d and e-1. Strong tractability means that the bound does not depend on d and depends polynomially on e-1. We prove that strong tractability holds iff ∑j=1∞ γj < ∞, where α measures the decay of Fourier coefficients in the weighted Korobov class, then for d ≥ 1, 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∞ γj1/2 < ∞, then for d ≥ 1, 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 γj < ∞. Hence, in particular, for γj = 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. © 2001 Academic Press.