Least-Squares Padé approximation of parametric and stochastic Helmholtz maps
For engineers and scientists solving parametric Helmholtz equations, this method offers a computationally efficient approximation, though it is an incremental extension of existing Padé techniques.
The paper introduces a rational model order reduction method based on single-point Least-Squares Padé approximation for parametric and stochastic Helmholtz maps, demonstrating its effectiveness on 2D numerical tests including transmission/reflection, scattering, and high-frequency problems, as well as stochastic wavenumber cases.
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.