TY  - JOUR
A1  - Ayala, Rafael Ceja
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
A1  - Pallikarakis, Nikolaos
T1  - Analysis of the transmission eigenvalue problem with two conductivity parameters
JF  - Applicable Analysis
N2  - In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work.
KW  - Transmission Eigenvalues
KW  - Conductive Boundary Condition
KW  - Inverse Scattering
Y1  - 2023
U6  - https://doi.org/10.1080/00036811.2023.2181167
SN  - 0003-6811
PB  - Taylor & Francis
ER  - 
TY  - JOUR
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
T1  - The inverse scattering problem for a conductive boundary condition and transmission eigenvalues
JF  - Applicable Analysis
N2  - In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside–outside duality method can be used to reconstruct the interior conductive eigenvalues.
KW  - Transmission eigenvalues
KW  - Conductive boundary condition
KW  - Inverse scattering
Y1  - 2018
U6  - https://doi.org/10.1080/00036811.2018.1504028
SN  - 1563-504X
VL  - 99
IS  - 3
SP  - 508
EP  - 529
PB  - Taylor & Francis
CY  - London
ER  - 
TY  - JOUR
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
T1  - Analysis and computation of the transmission eigenvalues with a conductive boundary condition
JF  - Applicable Analysis
N2  - We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber–Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown.
KW  - Boundary integral equations
KW  - Inverse spectral problem
KW  - Conductive boundary condition
KW  - Transmission eigenvalues
Y1  - 2020
U6  - https://doi.org/10.1080/00036811.2020.1789598
SN  - 1563-504X
VL  - 101
IS  - 6
SP  - 1880
EP  - 1895
PB  - Taylor & Francis
CY  - London
ER  - 
TY  - JOUR
A1  - Ayala, Rafael Ceja
A1  - Harris, Isaac
A1  - Kleefeld, Andreas
T1  - Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary
JF  - Inverse Problems and Imaging
N2  - In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.
Y1  - 2024
U6  - https://doi.org/10.3934/ipi.2023051
SN  - 1930-8337
SN  - 1930-8345 (eISSN)
VL  - 18
IS  - 3
SP  - 708
EP  - 729
PB  - AIMS
CY  - Springfield
ER  -