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

TL;DR

This paper proves the well-posedness of Bayesian shape inversion in acoustic scattering, explicitly showing how stability depends on wavenumber and excluding quasi-resonant regimes, advancing understanding of inverse problems at different frequencies.

Contribution

It establishes the wavenumber-explicit well-posedness of Bayesian shape inversion for acoustic scattering, a novel theoretical result in inverse problem analysis.

Findings

Well-posedness depends explicitly on wavenumber.

Stability is affected by frequency relative to spatial scale.

Results exclude quasi-resonant regimes.

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.