Inverse Scattering Theory and Transmission Eigenvalues

Cakoni, Fioralba; Colton, David; Haddar, Houssem

doi:10.1137/1.9781611974461

Cited by 181 publications

(346 citation statements)

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“…Transmission eigenvalue problems have been a very active field of research in the theory of inverse scattering. See the manuscripts [9] for a detailed account of the main results and techniques for these eigenvalue problems for the scalar scattering problem. First, we will study the problem for the electromagnetic scattering that is analogous to the problem studied in [6] and [20].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of two transmission eigenvalue problems with a coated boundary condition

Harris

2019

Applicable Analysis

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In this paper, we investigate two transmission eigenvalue problems associated with the scattering of a media with a coated boundary. In recent years, there has been a lot of interest in studying these eigenvalue problems. It can be shown that the eigenvalues can be recovered from the scattering data and hold information about the material properties of the media one wishes to determine. Motivated by recent works we will study the electromagnetic transmission eigenvalue problem and scalar 'zero-index' transmission eigenvalue problem for a media with a coated boundary. Existence of infinitely many real eigenvalues will be proven as well as showing that the eigenvalues depend monotonically on the refractive index and boundary parameter. Numerical examples in two spatial dimensions are presented for the scalar 'zero-index' transmission eigenvalue problem. Also, in our investigation we prove that as the boundary parameter tends to zero and infinity we recover the classical eigenvalue problems.

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Section: Introductionmentioning

confidence: 99%

“…In general, these eigenvalues can determined from the scattering data and can be used to determine information about the underline scattering object (see for e.g. [5], [9], [16], [19], [22], and [27]). In [10] the transmission eigenvalues are used to estimate the material properties of a highly oscillatory periodic scatterer.…”

Section: Introductionmentioning

confidence: 99%

Analysis of two transmission eigenvalue problems with a coated boundary condition

Harris

2019

Applicable Analysis

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“…This relationship is an equivalence provided m has compact support in R 2 , and all of the appropriate solutions are smooth enough (Cakoni et al, 2016). In particular, if m has support D, we notice that based on the formulation of (2), once we know the solution u on D, we can then extend the solution to any point in R 2 by evaluation of the integral.…”

Section: Lippmann-schwinger Equationmentioning

confidence: 97%

“…The Lippmann‐Schwinger equation is an integral formulation of the Helmholtz equation when used to analyze scattering problems. A solution

u

\begin{matrix} right Δ u + k^{2} n (x) u & left = 0 R^{2} & right \\ right u = u^{s} + u^{i} & left \\ right \lim_{r \to \infty} \sqrt{r} (\frac{\partial u^{s}}{\partial r} - i k u^{s}) & left = 0, \end{matrix}

is a solution to the integral equation

u false(x false) = u^{i} false(x false) - k^{2} \int_{{double-struckR}^{2}} normalΦ false(x, y false) m false(y false) u false(y false) normald y x \in {double-struckR}^{2},

where

m = 1 - n

and

normalΦ false(x, y false) = \frac{i}{4} H_{0}^{1} false(k false| x - y false| false)

is the fundamental solution to the background Helmholtz equation

Δ u + k^{2} u = 0 .

This relationship is an equivalence provided

m

has compact support in

{double-struckR}^{2}

, and all of the appropriate solutions are smooth enough (Cakoni et al, ). In particular, if

m

has support

D

, we notice that based on the formulation of , once we know the solution

…”

Section: Numerical Implementationmentioning

confidence: 99%

Object Identification in Radar Imaging via the Reciprocity Gap Method

Charnley

Wood

2020

Radio Science

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In this paper, we present an experimental method for locating and identifying objects in radar imaging, specifically problems that could arise in physical situations. The data for the forward problem are generated using a discretization of the Lippmann-Schwinger equation, and the inverse problem of object location is solved using the reciprocity gap approach to the linear sampling method. The main new development in this paper is an exploration of determining the permittivity of the object from the back-scattered data, utilizing another discretization of the Lippmann-Schwinger equation.

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“…However, this study is restricted to the case where the sought-after potentials are periodic. We must point out that most of the recent studies provide only theoretical results via dierent mathematical approaches (see, e. g., [35] [43]). The main goal of this paper is to study from both mathematical and numerical viewpoints the problem of determining refractive index proles from some measured or desired guided waves propagating in optical bers.…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis and solution methodology for an inverse spectral problem arising in the design of optical waveguides

Barucq

Bekkey

Djellouli

2018

Inverse Problems in Science and Engineering

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We analyze mathematically the problem of determining refractive index proles from some desired/measured guided waves propagating in optical bers. We establish the uniqueness of the solution of this inverse spectral problem assuming that only one guided mode is known. We then propose an iterative computational procedure for solving numerically the considered inverse spectral problem. Numerical results are presented to illustrate the potential of * corresponding author: rabia.djellouli@csun.edu 1 the proposed regularized Newton algorithm to eciently and accurately retrieve the refractive index proles even when the guided mode measurements are highly noisy.

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Inverse Scattering Theory and Transmission Eigenvalues

Abstract: All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics,

Cited by 181 publications

References 0 publications

Analysis of two transmission eigenvalue problems with a coated boundary condition

Analysis of two transmission eigenvalue problems with a coated boundary condition

Object Identification in Radar Imaging via the Reciprocity Gap Method

Mathematical analysis and solution methodology for an inverse spectral problem arising in the design of optical waveguides

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