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Generalized Sparse Grid Interpolation Based on the Fast Discrete Fourier Transform

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Book cover Sparse Grids and Applications - Munich 2018

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 144))

Abstract

In (Griebel and Hamaekers, Fast discrete Fourier transform on generalized sparse grids, Sparse grids and Applications, volume 97 of Lecture Notes in Computational Science and Engineering, pages 75–108, Springer, 2014), an algorithm for trigonometric interpolation involving only so-called standard information of multivariate functions on generalized sparse grids has been suggested and a study on its application for the interpolation of functions in periodic Sobolev spaces of dominating mixed smoothness has been presented. In this complementary paper, we now give a slight modification of the proofs, which yields an extension from the pairing \((\mathcal {H}^{s},\mathcal {H}_{{mix}}^t)\) to the more general pairing \((\mathcal {H}^{s},\mathcal {H}_{{mix}}^{t,r})\) and which in addition results in an improved estimate for the interpolation error. The improved (constructive) upper bound is in particular consistent with the lower bound for sampling on regular sparse grids with r = 0 and s = 0 given in (Dũng, Acta Mathematica Vietnamica, 43(1):83–110, 2018; Dũng et al., Hyperbolic Cross Approximation, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser/Springer, 2018).

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Acknowledgements

This work was supported in part by the Collaborative Research Centre 1060 of the Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. Institute for Numerical Simulation, University of Bonn, Bonn, Germany

    Michael Griebel

  2. Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven, Sankt Augustin, Germany

    Michael Griebel & Jan Hamaekers

Authors
  1. Michael Griebel
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  2. Jan Hamaekers
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Correspondence to Jan Hamaekers .

Editor information

Editors and Affiliations

  1. Institut für Informatik, TU München, Garching, Germany

    Hans-Joachim Bungartz

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Jochen Garcke

  3. Institut für Parallele und Verteilte Systeme (IPVS), Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

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Griebel, M., Hamaekers, J. (2021). Generalized Sparse Grid Interpolation Based on the Fast Discrete Fourier Transform. In: Bungartz, HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. Lecture Notes in Computational Science and Engineering, vol 144. Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_3

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