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Tight Tractability Results for a Model Second-Order Neumann Problem

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Abstract

We study the worst case complexity and the tractability of the model second-order problem \(-\Delta u+u=f\) on \(I^d=(0,1)^d \) with homogeneous Neumann boundary conditions. As is often the case, we study the variational formulation of this problem. Previous work on the tractability of problems such as this relied on the fact that such problems were reducible to the \(L_2(I^d)\)-approximation problem, which allowed us to find necessary and sufficient conditions for the problem to have a given degree of tractability. However, such an approach can only yield sufficient conditions, and not necessary conditions, for the model second-order Neumann problem to have a given degree of tractability. In this paper, we remedy this gap and find necessary and sufficient conditions for this Neumann problem to exhibit a given degree of tractability.

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Notes

  1. The definitions in this section would also work when \(H_d\) is a normed space rather than a Hilbert space, but since the specific space \(H_d\) we use is a Hilbert space, there is no need to introduce this extra generality.

  2. Note that the information \(N\) is nonadaptive. There is no need to consider adaptive information in this paper since our problem is linear, and it is well known that adaption does not help much for linear problems.

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Acknowledgments

We are grateful to the referees for their helpful suggestions, especially the connection of our problem to the approximation problem in the Besov and Triebel–Lizorkin norms. This research was supported in part by the National Science Foundation.

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Authors and Affiliations

  1. Department of Computer and Information Sciences, Fordham University, New York, NY , 10023, USA

    A. G. Werschulz

  2. Department of Computer Science, Columbia University, New York, NY , 10027, USA

    A. G. Werschulz & H. Woźniakowski

  3. Institute of Applied Mathematics, University of Warsaw, Warszawa, Poland

    H. Woźniakowski

Authors
  1. A. G. Werschulz
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  2. H. Woźniakowski
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Correspondence to A. G. Werschulz.

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Communicated by Phillipe G. Ciarlet.

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Werschulz, A.G., Woźniakowski, H. Tight Tractability Results for a Model Second-Order Neumann Problem. Found Comput Math 15, 899–929 (2015). https://doi.org/10.1007/s10208-014-9195-y

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  • DOI: https://doi.org/10.1007/s10208-014-9195-y

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