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Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part II

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • Erich Novak

    (Jena University, Math Institute)

  • Henryk Woźniakowski

    (University of Warsaw, Institute of Applied Mathematics and Mechanics
    Columbia University, Department of Computer Science)

Abstract

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1. the minimal errors for the univariate case decay polynomially fast to zero, 2. the largest singular value for the univariate case is simple and 3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.

Suggested Citation

  • Erich Novak & Henryk Woźniakowski, 2018. "Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part II," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 963-977, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_42
    DOI: 10.1007/978-3-319-72456-0_42
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