(1) School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China
Abstract:
We determine the exact order of -complexity of the numerical integration problem for the anisotropic class Wr(Id) and Hr(Id) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions Wp(Id) and Hp(Id). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration.
Project supported by the fund of Personnel Division of Nankai University and the Program of One Hundred Distinguished Chinese Scientists of the Chinese Academy of Sciences.