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Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

by R. Hiptmair and Ch. Schwab and E.A. Spence

(Report number 2024-21)

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 piece-wise 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 permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.

Keywords: frequency-domain acoustic scattering, Helmholtz equation, shape holomorphy, finite elements, perfectly matched layers, Smolyak quadrature, uncertainty quantification

@Techreport{HSS24_1103,
  author = {R. Hiptmair and Ch. Schwab and E.A. Spence},
  title = {Frequency-Explicit Shape Holomorphy
in Uncertainty Quantification for Acoustic Scattering},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2024-21},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2024/2024-21.pdf },
  year = {2024}
}

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