Least-Squares Pad\'e approximation of parametric and stochastic Helmholtz maps

TL;DR

This paper introduces a rational model order reduction method using Least-Squares Padé approximation for parametric and stochastic Helmholtz maps, demonstrating its effectiveness through numerical tests on wave problems.

Contribution

It develops a novel LS-Padé approximation technique for Helmholtz maps, including stochastic cases, with a new eigenvector-based computation of the Padé denominator.

Findings

Effective approximation of frequency response maps in wave problems

Weak convergence of measures in stochastic Helmholtz equations

Numerical tests confirm the method's accuracy and efficiency

Abstract

The present work deals with the rational model order reduction method based on the single-point Least-Square (LS) Pad\'e approximation technique introduced in [3]. Algorithmical aspects concerning the construction of the rational LS-Pad\'e approximant are described. In particular, the computation of the Pad\'e denominator is reduced to the calculation of the eigenvector corresponding to the minimal eigenvalue of a Gramian matrix. The LS-Pad\'e 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-Pad\'e approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation method.