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Huygens’ principle and iterative methods in inverse obstacle scattering
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  • Published: 22 July 2009

Huygens’ principle and iterative methods in inverse obstacle scattering

  • Olha Ivanyshyn1,
  • Rainer Kress1 &
  • Pedro Serranho2,3 

Advances in Computational Mathematics volume 33, pages 413–429 (2010)Cite this article

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Abstract

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083, Göttingen, Germany

    Olha Ivanyshyn & Rainer Kress

  2. CEMAT, Instituto Superior Técnico, Lisbon, Portugal

    Pedro Serranho

  3. Polytechnical Institute of Leiria, 2411-901, Leiria, Portugal

    Pedro Serranho

Authors
  1. Olha Ivanyshyn
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  2. Rainer Kress
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  3. Pedro Serranho
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Corresponding author

Correspondence to Rainer Kress.

Additional information

Communicated by the guest editors Benny Hon, Jin Cheng and Masahiro Yamamoto.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Ivanyshyn, O., Kress, R. & Serranho, P. Huygens’ principle and iterative methods in inverse obstacle scattering. Adv Comput Math 33, 413–429 (2010). https://doi.org/10.1007/s10444-009-9135-6

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  • Received: 15 April 2008

  • Accepted: 22 June 2009

  • Published: 22 July 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s10444-009-9135-6

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Keywords

  • Inverse scattering
  • Nonlinear integral equations
  • Sound-soft obstacle

Mathematics Subject Classifications (2000)

  • 35J05
  • 35R25
  • 35R30
  • 45Q05
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