Frequency‐dependent contraction rates for the Bayesian method to the inverse source problem

This paper addresses an inverse source problem for acoustic waves in a range of frequencies. Our study has two main goals. First, although the problem is severely ill‐posed with a logarithmic stability estimate, we demonstrate, through careful analysis of the forward map's singular values, that increasing the frequency range enhances stability, a phenomenon known as increasing resolution/stability. Second, motivated by the singular value behavior, we establish a consistency theorem for the inverse problem using a nonparametric Bayesian approach. Despite ill‐posedness, we prove that the posterior distribution contracts around the true source at a rate combining polynomial and logarithmic terms, both dependent on the frequency range, reflecting the increasing resolution/stability phenomena in a Bayesian framework. These results also provide practical guidance for balancing quantity, quality, and cost of measurement.