A Bayesian approach to inverse problems in spaces of measures
Phuoc-Truong Huynh
TL;DR
This work develops a Bayesian framework for inverse problems where the unknown parameter is a vector-valued Radon measure, enabling sparsity via measure-valued priors. It defines priors on the space of measures through vector-valued point processes, analyzes posterior well-posedness in the weak* topology, and proves stability of the posterior with respect to data perturbations, as well as convergence of posterior approximations under discretized forward models. The main contributions include a rigorous construction of priors on measure spaces, a Lévy-Khintchine type characterization of random measures, and general well-posedness and approximation results for the posterior, with applications to Gaussian noise settings and physically motivated examples such as Gaussian convolution and Helmholtz source localization. The approach provides a principled, sparse, infinite-dimensional framework for Bayesian inverse problems with practical implications for applications in imaging, acoustics, and geophysics, where localized or discrete sources are common.
Abstract
In this work, we develop a Bayesian framework for solving inverse problems in which the unknown parameter belongs to a space of Radon measures taking values in a separable Hilbert space. The inherent ill-posedness of such problems is addressed by introducing suitable measure-valued priors that encode prior information and promote desired sparsity properties of the parameter. Under appropriate assumptions on the forward operator and noise model, we establish the well-posedness of the Bayesian formulation by proving the existence, uniqueness, and stability of the posterior with respect to perturbations in the observed data. In addition, we also discuss computational strategies for approximating the posterior distribution. Finally, we present some examples that demonstrate the effectiveness of the proposed approach.