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Wavenumber-Explicit Well-Posedness of Bayesian Shape Inversion in Acoustic Scattering

Safiere Kuijpers, Laura Scarabosio

Abstract

We consider the Bayesian approach to the inverse problem of recovering the shape of an object from measurements of its scattered acoustic field. Working in the time-harmonic setting, we focus on a Helmholtz transmission problem and then extend our results to an exterior Dirichlet problem. We assume the scatterer to be star-shaped and we use, as prior, a truncated expansion with uniform random variables for a radial parametrization of the scatterer's boundary. The main novelty of our work is that we establish the well-posedness of the Bayesian shape inverse problem in a wavenumber-explicit way, under some conditions on the material parameters excluding quasi-resonant regimes. Our estimates highlight how the stability of the posterior with respect to the data is affected by the wavenumber (or, in other words, the frequency), whose magnitude has to be understood not in absolute terms but in relationship to the spatial scale of the problem.

Wavenumber-Explicit Well-Posedness of Bayesian Shape Inversion in Acoustic Scattering

Abstract

We consider the Bayesian approach to the inverse problem of recovering the shape of an object from measurements of its scattered acoustic field. Working in the time-harmonic setting, we focus on a Helmholtz transmission problem and then extend our results to an exterior Dirichlet problem. We assume the scatterer to be star-shaped and we use, as prior, a truncated expansion with uniform random variables for a radial parametrization of the scatterer's boundary. The main novelty of our work is that we establish the well-posedness of the Bayesian shape inverse problem in a wavenumber-explicit way, under some conditions on the material parameters excluding quasi-resonant regimes. Our estimates highlight how the stability of the posterior with respect to the data is affected by the wavenumber (or, in other words, the frequency), whose magnitude has to be understood not in absolute terms but in relationship to the spatial scale of the problem.

Paper Structure

This paper contains 12 sections, 15 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model Problem
  3. Shape Modeling
  4. Helmholtz Scattering Transmission Problem
  5. Bayesian Inverse Problem Setting
  6. Definition of the Bayesian Shape Inverse Problem
  7. General Well-Posedness Results
  8. Wavenumber-Explicit Well-Posedness
  9. Preliminaries on Wavenumber-Explicit Bounds
  10. Well-Posedness of Bayesian Shape Inversion
  11. Extension to Sound-Soft Obstacle Scattering
  12. Conclusions

Key Result

Lemma 2.1

If Assumptions ass:Expansion-ass:uniform hold, then there exists $\tilde{\gamma}>0$ such that, for $\mathbb{P}$-a.e. $\omega\in\Omega$, $D_{in}(\omega)$ is star-shaped with respect to a ball of radius $\tilde{\gamma}\mathrm{diam}(D_{in}(\omega))$. For $d=2$, we can take $\tilde{\gamma}=\left(\frac{(

Figures (1)

  • Figure 1: Description of the physical setting (in two space dimensions): an incoming plane wave scatters on a homogeneous, star-shaped particle.

Theorems & Definitions (26)

  • Remark 2.1: Radius regularity
  • Remark 2.2: On finite dimensional noise
  • Lemma 2.1
  • proof
  • Definition 2.1
  • Theorem 2.1
  • Proposition 3.1: Theorem 6.31 in stuart2010inverse
  • Proposition 3.2: Proposition 3 in hoang2012sparse
  • Corollary 3.1
  • proof
  • ...and 16 more