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 (sound-soft) problem. It is well known that higher frequencies yield higher resolution but greater sensitivity to noise; here we give the first rigorous results quantifying how this sensitivity to noise depends on the wavenumber. We model the scatterer as star-shaped, with a prior on its boundary given by a series expansion of the angle-dependent radius with uniformly distributed coefficients. Our main result establishes well-posedness of the Bayesian shape inverse problem with constants explicit in the wavenumber, under problem-specific conditions on the material parameters that exclude quasi-resonant regimes. Stability estimates in the Hellinger and 1-Wasserstein metrics show that the stability constant of the posterior with respect to the data grows exponentially with the square of the wavenumber, whose magnitude must be understood not in absolute terms but relative to the spatial scale of the problem. Numerical experiments illustrate this effect.
From: Laura Scarabosio [view email]
[v1]
Wed, 30 Oct 2024 15:12:29 UTC (72 KB)
[v2]
Sun, 23 Aug 2026 17:18:18 UTC (1,704 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。