American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering
  • IPI Home
  • This Issue
  • Next Article
    On a Hybrid method for shape reconstruction of a buried object in an elastostatic half plane
May  2009, 3(2): 211-229. doi: 10.3934/ipi.2009.3.211

The inverse acoustic obstacle scattering problem and its interior dual

Brian Sleeman 1,

1. 

School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

Received  November 2008 Revised  March 2009 Published  May 2009

This paper addresses possible connections between two classical inverse problems arising in wave propagation. The first is the problem of extracting geometrical information about an unknown bounded domain from a knowledge its eigen-frequencies. The chosen method of investigation being the high frequency asymptotics of the associated counting function. The second problem is the inverse obstacle scattering problem. That is the determination of an unknown obstacle from far field data. This problem is investigated through the high frequency asymptotics of the associated scattering phase. It turns out that there is a remarkable similarity between the asymptotic expansions in each of these problems. We discuss a number of ideas and techniques along the way including representations of the scattering matrix and the Kirchoff approximation. We also show how to solve scattering problems for polygonal obstacles. Whether there is a deep physical connection between interior and exterior scattering problems remains a challenging area of research.
Keywords: Eigenvalue problem, acoustic obstacle scattering, isospectrality, scattering matrix, scattering phase..
Mathematics Subject Classification: Primary:35P20, 78A46;Secondary 34E20, 78A45, 81U2.
Citation: Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[1]

Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems & Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749
[2]

Giorgio Menegatti, Luca Rondi. Stability for the acoustic scattering problem for sound-hard scatterers. Inverse Problems & Imaging, 2013, 7 (4) : 1307-1329. doi: 10.3934/ipi.2013.7.1307
[3]

Yi-Hsuan Lin. Reconstruction of penetrable obstacles in the anisotropic acoustic scattering. Inverse Problems & Imaging, 2016, 10 (3) : 765-780. doi: 10.3934/ipi.2016020
[4]

Fenglong Qu, Jiaqing Yang. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 281-291. doi: 10.3934/ipi.2018012
[5]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335
[6]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems & Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
[7]

Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026
[8]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[9]

Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995
[10]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[11]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455
[12]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[13]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[14]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[15]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617
[16]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039
[17]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131
[18]

Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951
[19]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[20]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences