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  - http://dx.doi.org/10.1080/00036811.2023.2181167
SN  - 0003-6811
PB  - Taylor & Francis
ER  - 
TY  - JOUR
A1  - Grajewski, Matthias
A1  - Kleefeld, Andreas
T1  - Detecting and approximating decision boundaries in low-dimensional spaces
JF  - Numerical Algorithms
N2  - A method for detecting and approximating fault lines or surfaces, respectively, or decision curves in two and three dimensions with guaranteed accuracy is presented. Reformulated as a classification problem, our method starts from a set of scattered points along with the corresponding classification algorithm to construct a representation of a decision curve by points with prescribed maximal distance to the true decision curve. Hereby, our algorithm ensures that the representing point set covers the decision curve in its entire extent and features local refinement based on the geometric properties of the decision curve. We demonstrate applications of our method to problems related to the detection of faults, to multi-criteria decision aid and, in combination with Kirsch’s factorization method, to solving an inverse acoustic scattering problem. In all applications we considered in this work, our method requires significantly less pointwise classifications than previously employed algorithms.
KW  - MCDA
KW  - Inverse scattering problem
KW  - Fault approximation
KW  - Fault detection
Y1  - 2023
SN  - 1572-9265
N1  - Corresponding author: Matthias Grajewski
VL  - 93
IS  - 4
PB  - Springer Science+Business Media
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Pieronek, Lukas
A1  - Kleefeld, Andreas
T1  - On trajectories of complex-valued interior transmission eigenvalues
JF  - Inverse problems and imaging : IPI
N2  - This paper investigates the interior transmission problem for homogeneous media via eigenvalue trajectories parameterized by the magnitude of the refractive index. In the case that the scatterer is the unit disk, we prove that there is a one-to-one correspondence between complex-valued interior transmission eigenvalue trajectories and Dirichlet eigenvalues of the Laplacian which turn out to be exactly the trajectorial limit points as the refractive index tends to infinity. For general simply-connected scatterers in two or three dimensions, a corresponding relation is still open, but further theoretical results and numerical studies indicate a similar connection.
KW  - Interior transmission problem
KW  - Eigenvalue trajectories
KW  - Complex-valued eigenvalues
Y1  - 2024
U6  - http://dx.doi.org/10.3934/ipi.2023041
SN  - 1930-8337 (Print)
SN  - 1930-8345 (Online)
VL  - 18
IS  - 2
SP  - 480
EP  - 516
PB  - AIMS
CY  - Springfield, Mo
ER  -