Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

TL;DR

This paper establishes frequency-explicit holomorphic dependence of the scattered acoustic field on uncertain shape parameters, enabling efficient surrogate modeling and quantification of discretization effects in high-frequency acoustic scattering problems.

Contribution

It provides frequency-explicit holomorphicity results for the scattered field with uncertain shapes, incorporating PML effects and finite-element discretization analysis.

Findings

Frequency-explicit holomorphic dependence of the scattered field on shape parameters.

Quantitative estimates for finite-element discretization impact.

Framework for high-dimensional surrogate model construction.

Abstract

We consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable obstacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piecewise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Mathematical Models and Methods in Applied Sciences, 29 (2019), pp. 317-354] we obtain frequency-explicit statements on the holomorphic dependence of the scattered field and the far-field pattern on the stochastic shape parameters. This paves the way for applying general results on the efficient construction of high-dimensional surrogate models. We also take into account the effect of domain truncation by means of perfectly matched layers (PML). In addition, spatial regularity estimates which are explicit in terms of the wavenumber $k$ permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.