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Error Estimates for Approximate Operator Inversion via Kernel-Based Methods

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Book cover Curves and Surfaces (Curves and Surfaces 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

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Abstract

In this paper we investigate error estimates for the approximate solution of operator equations \(Af=u\), where u needs not to be a function on the same domain as f. We use the well-established theory of generalized interpolation, also known as optimal recovery in reproducing kernel Hilbert spaces, to generate an approximation to f from finitely many samples \(u(x_1), \dots , u(x_N)\). To derive error estimates for this approximation process we will show sampling inequalities on fairly general Riemannian manifolds.

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Acknowledgement

The author would like to thank Grady B. Wright for the possibility to gave a talk in the minisymposium ‘Kernel Based Approximation Methods’ at the conference ‘Curves and Surfaces’ 2014. Moreover he would like to thank Frank Filbir for fruitful discussions on the topic and careful proofreading.

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Authors and Affiliations

  1. Faculty of Mathematics, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Kristof Schröder

  2. Helmholtz Centre Munich, Ingolstädter Landstraße 1, 85764, Neuherberg, Germany

    Kristof Schröder

Authors
  1. Kristof Schröder
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Correspondence to Kristof Schröder .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Schröder, K. (2015). Error Estimates for Approximate Operator Inversion via Kernel-Based Methods. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_28

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  • DOI: https://doi.org/10.1007/978-3-319-22804-4_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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