Interpolatory Rational Model Order Reduction of Parametric Problems Lacking Uniform Inf-Sup Stability

We present a technique for the approximation of a class of Hilbert space--valued maps which arise within the framework of model order reduction (MOR) for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR strategy consists in constructing an explicit rational approximation based on few snapshots of the solution in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g., resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique.