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Reconstruction and Collocation of a Class of Non-periodic Functions by Sampling Along Tent-Transformed Rank-1 Lattices

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Abstract

Spectral collocation and reconstruction methods have been widely studied for periodic functions using Fourier expansions. We investigate the use of cosine series for the approximation and collocation of multivariate non-periodic functions with frequency support mainly determined by hyperbolic crosses. We seek methods that work for an arbitrary number of dimensions. We show that applying the tent-transformation on rank-1 lattice points renders them suitable to be collocation/sampling points for the approximation of non-periodic functions with perfect numerical stability. Moreover, we show that the approximation degree—in the sense of approximating inner products of basis functions up to a certain degree exactly—of the tent-transformed lattice point set with respect to cosine series, is the same as the approximation degree of the original lattice point set with respect to Fourier series, although the error can still be reduced in the case of cosine series. A component-by-component algorithm is studied to construct such a point set. We are then able to reconstruct a non-periodic function from its samples and approximate the solutions to certain PDEs subject to Neumann and Dirichlet boundary conditions. Finally, we present some numerical results.

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Acknowledgments

We thank Daan Huybrechs for his comments on the initial transcripts of the paper. We thank Dirk Abbeloos for some of his valuable tips on PDEs. We also thank our anonymous referees for their constructive review comments. We thank the ICERM organization at Brown University for their kind hospitality during the semester program for High-dimensional Approximation, many ideas to improve this manuscript came while working in the vast collaborative spaces they provided.

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Authors and Affiliations

  1. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001, Heverlee, Belgium

    Gowri Suryanarayana, Dirk Nuyens & Ronald Cools

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  1. Gowri Suryanarayana
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  2. Dirk Nuyens
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  3. Ronald Cools
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Correspondence to Gowri Suryanarayana.

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Communicated by Arieh Iserles.

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Suryanarayana, G., Nuyens, D. & Cools, R. Reconstruction and Collocation of a Class of Non-periodic Functions by Sampling Along Tent-Transformed Rank-1 Lattices. J Fourier Anal Appl 22, 187–214 (2016). https://doi.org/10.1007/s00041-015-9412-3

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  • DOI: https://doi.org/10.1007/s00041-015-9412-3

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