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Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

Ralf Hiptmair, Christoph Schwab, Euan A. Spence

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.

Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

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 permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.
Paper Structure (26 sections, 28 theorems, 168 equations, 1 figure)

This paper contains 26 sections, 28 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scattering Transmission Problem
  3. Uncertainty Quantification (UQ) via Polynomial Surrogate Models
  4. Simplest Case: Size Uncertainty Quantification
  5. Scattering problems without quasi-resonances
  6. Layout of the paper
  7. Variational Formulation of the Transmission Problem with Parametric Interface
  8. A parametric family of diffeomorphisms
  9. Variational formulation under pullback via the domain mapping $\Phi$
  10. $k$-Explicit Norm Bounds for the Solution of the Transmission Problem
  11. $k$-Explicit Holomorphic Dependence of the Solution on the Radial Displacement Function
  12. Recap of $k$-explicit Finite-Element Error Bounds
  13. Definition of radial PML truncation.
  14. Properties of the sesquilinear form and solution operator of the PML problem
  15. The accuracy of PML truncation for $k$ large.
  16. ...and 11 more sections

Key Result

Lemma 2.2

Under Assumption ass:rbd and chi the mappings $\boldsymbol{\Phi}(r;\cdot)$, $r\in{\cal R}$, as defined in eq:Phi are diffeomorphisms of class $C^{m-1,1}$, map the unit sphere $\partial \widehat{D} = \mathbb{S}^{d-1}$ onto the interface $\Gamma(r)$, and agree with the identity in the exterior of the

Figures (1)

  • Figure 1.1: Dependence of norms ${\left\|{u}\right\|_{L^{2}(B_{2})}}$ and ${\left\|{u}\right\|_{H^{1}(B_{2})}}$ of the solution $u$ of the transmission problem \ref{['eq:htp']} on $k$, when $d=2$, $D$ is the unit disk, and $u_{\mathrm{inc}}({\boldsymbol{x}})=\exp(\rm i k x_{1})$. For $n_{i}=3$ quasi-resonances manifest themselves as spikes of the graph $k\mapsto\left\|{u}\right\|_{H^{1}(B_{2})}$, for $n_{i}=\frac{1}{3}$ such spikes are conspicuously absent. The norms were computed by means of a Fourier spectral method with a number of modes large enough to render any discretization error negligible. MATLAB codes in https://github.com/hiptmair/ScatteringQuasiResonances.

Theorems & Definitions (59)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.2: Properties of mappings $\boldsymbol{\Phi}$
  • Lemma 2.3: Jacobian of transformation mapping
  • proof
  • Lemma 2.4: Transformation of norms
  • Theorem 3.1: $k$-explicit bound on the solution of the transmission problem from MS19
  • proof : References for the proof
  • Remark 3.2: The Helmholtz exterior Dirichlet problem
  • Theorem 4.1: $k$-explicit condition for existence of $\widehat{u}(\mathfrak{r};\cdot)$ and related bounds
  • ...and 49 more