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 c 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 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 c j < ., and tractability holds iff lim sup d Q . ; d j=1 c 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 c 1/a j < ., where a measures the decay of Fourier coefficients in the weighted Korobov class, then for d \ 1, n prime and d > 0 there exist lattice rules that satisfy an error bound independent of d and of order n −a/2+d . This is almost the best possible result, since the order n −a/2 cannot be improved upon even for d=1. A corresponding result is deduced for weighted non-periodic Sobolev spaces: if ; there exist shifted lattice rules that satisfy an error bound independent of d and of order n −1+d . We also check how the randomized error of the (classical

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