@article{AyalaHarrisKleefeldetal.2023,
  author    = {Ayala, Rafael Ceja and Harris, Isaac and Kleefeld, Andreas and Pallikarakis, Nikolaos},
  title     = {Analysis of the transmission eigenvalue problem with two conductivity parameters},
  series = {Applicable Analysis},
  journal   = {Applicable Analysis},
  publisher = {Taylor \& Francis},
  issn      = {0003-6811},
  doi       = {10.1080/00036811.2023.2181167},
  pages     = {37 Seiten},
  year      = {2023},
  abstract  = {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.},
  language  = {en}
}
@article{GrajewskiKleefeld2023,
  author    = {Grajewski, Matthias and Kleefeld, Andreas},
  title     = {Detecting and approximating decision boundaries in low-dimensional spaces},
  series = {Numerical Algorithms},
  volume    = {93},
  journal   = {Numerical Algorithms},
  number    = {4},
  publisher = {Springer Science+Business Media},
  address   = {Dordrecht},
  issn      = {1572-9265},
  pages     = {35 Seiten},
  year      = {2023},
  abstract  = {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.},
  language  = {en}
}
@article{PieronekKleefeld2024,
  author    = {Pieronek, Lukas and Kleefeld, Andreas},
  title     = {On trajectories of complex-valued interior transmission eigenvalues},
  series = {Inverse problems and imaging : IPI},
  volume    = {18},
  journal   = {Inverse problems and imaging : IPI},
  number    = {2},
  publisher = {AIMS},
  address   = {Springfield, Mo},
  issn      = {1930-8337 (Print)},
  doi       = {10.3934/ipi.2023041},
  pages     = {480 -- 516},
  year      = {2024},
  abstract  = {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.},
  language  = {en}
}