Least-Squares Padé approximation of parametric and stochastic Helmholtz maps

The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Pade approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261-1284 2018, Math. Comput. 89, 1229-1257 2020). Algorithmical aspects concerning the construction of rational LS-Pade approximants are described. In particular, we show that the computation of the Pade denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Pade techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both 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-Pade approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods.