Looking Back on Inverse Scattering Theory

We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves over the past fifty years together with some personal memories of our participation in these...

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[18]  Peter Monk,et al.  The linear sampling method for solving the electromagnetic inverse medium problem , 2002 .

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[20]  Rainer Kress,et al.  Inverse scattering from an open arc , 1995 .

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[23]  David Colton,et al.  The uniqueness of a solution to an inverse scattering problem for electromagnetic waves , 1992 .

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[32]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[33]  Jan Sokolowski,et al.  On the Topological Derivative in Shape Optimization , 1999 .

[34]  Gang Bao,et al.  Inverse Medium Scattering Problems for Electromagnetic Waves , 2005, SIAM J. Appl. Math..

[35]  G. Bao,et al.  Inverse medium scattering for the Helmholtz equation at fixed frequency , 2005 .

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[37]  Fioralba Cakoni,et al.  A UNIQUENESS THEOREM FOR AN INVERSE ELECTROMAGNETIC SCATTERING PROBLEM IN INHOMOGENEOUS ANISOTROPIC MEDIA , 2003, Proceedings of the Edinburgh Mathematical Society.

[38]  A. Kirsch.  Remarks on the Born approximation and the Factorization Method , 2017 .

[39]  Georgi Vodev,et al.  Transmission Eigenvalue-Free Regions , 2014, 1401.1627.

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[43]  Steven Kusiak,et al.  The scattering support , 2003 .

[44]  A. Bukhgeǐm,et al.  Recovering a potential from Cauchy data in the two-dimensional case , 2008 .

[45]  Rainer Kress,et al.  Newton’s method for inverse obstacle scattering meets the method of least squares , 2003 .

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[47]  David Colton,et al.  Dense sets and far field patterns for acoustic waves in an inhomogeneous medium , 1988, Proceedings of the Edinburgh Mathematical Society.

[48]  Fioralba Cakoni,et al.  Transmission Eigenvalues , 2021, Applied Mathematical Sciences.

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[50]  Armin Lechleiter,et al.  The linear sampling method revisited , 2009 .

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[55]  T. Arens,et al.  Why linear sampling works , 2004 .

[56]  Fioralba Cakoni,et al.  Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals , 2013 .

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[58]  Andreas Kirsch,et al.  The Denseness of the Far Field Patterns for the Transmission Problem , 1986 .

[59]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[60]  Georgi Vodev,et al.  High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues , 2017, 1701.04668.

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[62]  R. Kleinman,et al.  An inverse transmission problem for the Helmholtz equation , 1987 .

[63]  Evgeny Lakshtanov,et al.  Ellipticity in the Interior Transmission Problem in Anisotropic Media , 2012, SIAM J. Math. Anal..

[64]  Changmei Liu,et al.  Inverse obstacle problem: Local uniqueness for rougher obstacles and the identification of a ball , 1997 .

[65]  Roland Potthast,et al.  A fast new method to solve inverse scattering problems , 1996 .

[66]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[67]  Rainer Kress,et al.  On an Integral Equation of the first Kind in Inverse Acoustic Scattering , 1986 .

[68]  Rainer Kress,et al.  NONLINEAR INTEGRAL EQUATIONS IN INVERSE OBSTACLE SCATTERING , 2006 .

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[70]  P. M. Berg,et al.  A modified gradient method for two-dimensional problems in tomography , 1992 .

[71]  M. Burger.  A level set method for inverse problems , 2001 .

[72]  Roland Potthast,et al.  Frechet differentiability of boundary integral operators in inverse acoustic scattering , 1994 .

[73]  Andreas Kirsch,et al.  Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory , 1999 .

[74]  Jin Cheng,et al.  Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves , 2005 .

[75]  Lars Mönch,et al.  On the inverse acoustic scattering problem by an open arc: the sound-hard case , 1997 .

[76]  R. Novikov,et al.  Multidimensional inverse spectral problem for the equation —Δψ + (v(x) — Eu(x))ψ = 0 , 1988 .

[77]  A. Roger,et al.  Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem , 1981 .

[78]  M. Tadia,et al.  An inverse problem for Helmholtz equation , 2012 .

[79]  Thorsten Hohage,et al.  Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem , 2006, J. Comput. Phys..

[80]  Peter Monk,et al.  The Linear Sampling Method in Inverse Electromagnetic Scattering , 2010 .

[81]  J. Sylvester.  Notions of support for far fields , 2006 .

[82]  Thorsten Hohage,et al.  On the numerical solution of a three-dimensional inverse medium scattering problem , 2001 .

[83]  N I Grinberg,et al.  The Factorization Method for Inverse Problems , 2007 .

[84]  Victor Isakov,et al.  On uniqueness in th invese transmission scattering problem , 1990 .

[85]  John C. Schotland,et al.  Inverse Born Series for Diffuse Waves , 2009 .

[86]  Houssem Haddar,et al.  On the Fréchet Derivative for Obstacle Scattering with an Impedance Boundary Condition , 2004, SIAM J. Appl. Math..

[87]  J Blohbaum,et al.  Optimisation methods for an inverse problem with time-harmonic electromagnetic waves: an inverse problem in electromagnetic scattering , 1989 .

[88]  R. Kress,et al.  Nonlinear integral equations and the iterative solution for an inverse boundary value problem , 2005 .

[89]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[90]  Guanghui Hu,et al.  Shape Identification in Inverse Medium Scattering Problems with a Single Far-Field Pattern , 2015, SIAM J. Math. Anal..

[91]  A. Kirsch.  The domain derivative and two applications in inverse scattering theory , 1993 .