A Bayesian approach to sparse inverse problems

In this work, we study a Bayesian approach to the inverse problem of recovering an unknown sparse Radon measure u from a noisy observation y of its image through a known (possibly non-linear) forward G. We first define a prior measure on the space of Radon measures via point processes. We then study the well-posedness of the Bayesian inverse problem under a mild setting, which takes into account the above-mentioned prior measure. A sampling algorithm is proposed to compute an approximation of the associated posterior measure. Finally, we discuss a practical aspect of this Bayesian approach by considering the problem of identifying sound sources from pressure measurements. Here, the acoustics are modeled by the Helmholtz equation, and our goal is to get information about the strength and position of the sound sources.

This is joint work with Barbara Kaltenbacher and Alexander Mozdzen.