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.

翻译：本文研究基于贝叶斯方法从散射声场测量数据中重构物体形状的反问题。在时谐场设定下，我们首先聚焦于亥姆霍兹传输问题，随后将结果推广至外部狄利克雷问题。假设散射体呈星形结构，我们采用截断展开形式作为先验分布，其中散射体边界的径向参数化由均匀随机变量描述。本工作的主要创新在于：在排除准共振区域的材料参数条件下，我们以波数显式的方式建立了贝叶斯形状反演问题的适定性。我们的估计结果揭示了后验分布对数据的稳定性如何受波数（即频率）影响，其中波数量级需结合问题的空间尺度进行理解，而非仅考虑其绝对值。