Numerical methods for wave propagation phenomena are extremely computationally intensive in mid- and high-frequency regimes, since they demand high-order polynomial approximations and highly refined meshes. For this reason, a frequency response analysis in a large range of frequencies of interest has often a prohibitive computational cost. In this framework, Model Order Reduction (MOR) can be very effective: starting from few expensive evaluations of the frequency response map, a MOR method provides a reliable approximation of the frequency response function, which can be evaluated quite cheaply in a whole range of frequencies.
In this project, we focus on the minimal rational interpolation method, which approximates the frequency response map by rational functions. In reasonably general one-dimensional settings, one can prove convergence of the surrogate to the exact map as the number of offline samples increases. Moreover, starting from such theoretical results, adaptive algorithms based on a greedy strategy have been proposed, which allow to identify the ``optimal'' placement and number of samples to be taken, in order for a certain accuracy to be achieved.
Starting from these encouraging results, we face the problem of extending minimal rational interpolation to higher-dimensional parametric dependencies, for instance scattering problems over parametric/stochastic domains, and/or involving materials characterized by parametric/stochastic physical properties. Within this project, we wish to determine some properties of the resulting multivariate solution maps with respect to the extra parameters, which are somewhat unexplored. Then, we plan to investigate the possibility of employing minimal rational interpolation (possibly in its greedy version) for the high-dimensional MOR.