Andreas Kleefeld

Davide Pradovera, Alessandro Borghi, Lukas Pieronek et al.

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the corresponding spectral map from material parameters to eigenpairs may exhibit non-smooth or bifurcating behavior. In this work, we develop a theoretical framework identifying sufficient conditions for such non-smooth spectral behavior in the ITP on general domains. We further specialize our analysis to some radially symmetric geometries, enabling a more precise characterization of bifurcations in the spectrum. Computationally, we formulate the ITP as a parametric, discrete, nonlinear eigenproblem and use a match-based adaptive contour eigensolver to accurately and efficiently track eigenvalue trajectories under parameter variation. Numerical experiments confirm the theoretical predictions and reveal novel non-smooth spectral effects.

Marvin Kahra, Michael Breuß, Andreas Kleefeld et al.

Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric $2\times2$ matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.

Marvin Kahra, Michael Breuß, Andreas Kleefeld et al.

Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves challenging to define them generally for higher-dimensional data, such as colour representations. Numerous approaches have therefore been taken to solve this problem with certain compromises. In this paper we will analyse the construction of a new approach, which we have already presented experimentally in paper [Kahra, M., Breuß, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]. This is based on a method by Burgeth and Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254], who regard the colours as symmetric $2\times2$ matrices and compare them by means of the Loewner order in a bi-cone through different suprema. However, we will replace the supremum with the LogExp approximation for the maximum instead. This allows us to transfer the associativity of the dilation from the one-dimensional case to the higher-dimensional case. In addition, we will investigate the minimality property and specify a relaxation to ensure that our approach is continuously dependent on the input data.

Andreas Kleefeld

Shape optimization problems for interior eigenvalues is a very challenging task since already the computation of interior eigenvalues for a given shape is far from trivial. For example, a concrete maximizer with respect to shapes of fixed area is theoretically established only for the first two non-trivial Neumann eigenvalues. The existence of such a maximizer for higher Neumann eigenvalues is still unknown. Hence, the problem should be addressed numerically. Better numerical results are achieved for the maximization of some Neumann eigenvalues using boundary integral equations for a simplified parametrization of the boundary in combination with a non-linear eigenvalue solver. Shape optimization for interior transmission eigenvalues is even more complicated since the corresponding transmission problem is non-self-adjoint and non-elliptic. For the first time numerical results are presented for the minimization of interior transmission eigenvalues for which no single theoretical result is yet available.