Davide Pradovera

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.

Francesca Bonizzoni, Fabio Nobile, Ilaria Perugia et al.

The present work deals with the rational model order reduction method based on the single-point Least-Square (LS) Padé approximation technique introduced in [3]. Algorithmical aspects concerning the construction of the rational LS-Padé approximant are described. In particular, the computation of the Padé denominator is reduced to the calculation of the eigenvector corresponding to the minimal eigenvalue of a Gramian matrix. The LS-Padé technique is employed to approximate the frequency response map associated to various parametric time-harmonic wave problems, namely, a transmission/reflection problem, a scattering problem and a problem in high-frequency regime. In all cases we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution, and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation method.