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Computational complexity of the integration problem for anisotropic classes

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Abstract

We determine the exact order of ɛ-complexity of the numerical integration problem for the anisotropic class W r∞ (Id) and H r∞ (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 W Λp (Id) and H Λp (Id). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Ye Peixin

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  1. Ye Peixin
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Correspondence to Ye Peixin.

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Communicated by A. Zhou

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.

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Peixin, Y. Computational complexity of the integration problem for anisotropic classes. Adv Comput Math 23, 375–392 (2005). https://doi.org/10.1007/s10444-004-1830-8

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  • DOI: https://doi.org/10.1007/s10444-004-1830-8

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