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Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions

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Extraction of Quantifiable Information from Complex Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

Abstract

A straightforward discretisation of high-dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional functions: We present stable and effective algorithms for the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on an index set \(\mathcal{I}\subset \mathbb{Z}^{d}\).

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Acknowledgements

We gratefully acknowledge support by the German Research Foundation (DFG) within the Priority Program 1324, project PO 711/10-2 and KU 2557/1-2. Moreover, Ines Melzer and Stefan Kunis gratefully acknowledge their support by the Helmholtz Association within the young investigator group VH-NG-526.

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Authors and Affiliations

  1. Technical University of Chemnitz, Reichenhainer Str. 39, 09126, Chemnitz, Germany

    Lutz Kämmerer, Daniel Potts & Toni Volkmer

  2. University of Osnabrück, Albrechtstrasse 28a, 49069, Osnabrück, Germany

    Stefan Kunis & Ines Melzer

Authors
  1. Lutz Kämmerer
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  2. Stefan Kunis
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  3. Ines Melzer
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  4. Daniel Potts
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  5. Toni Volkmer
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Correspondence to Stefan Kunis .

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Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

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

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Klaus Ritter

  6. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

  8. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Harry Yserentant

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Kämmerer, L., Kunis, S., Melzer, I., Potts, D., Volkmer, T. (2014). Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_17

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