Inversion Bayésienne : illustration sur des problèmes tomographiques et astrophysiques

In this paper we provide an algorithm allowing to solve the variational Bayesian issue as a functional optimization problem. The main contribution of this paper is to transpose a classical iterative algorithm of optimization in the metric space of measures involved in the Bayesian methodology. Once given the convergence properties of this algorithm, we consider its application to large dimensional inverse problems, especially for unsupervised reconstruction. The interest of our algorithm is enhanced by its application to large dimensional linear inverse problems involving sparse objects. Finally, we provide simulation results. First we show the good numerical performances of our method comparatively with classical ones on a small example. On a second time we treat a large dimensional dictionary learning problem. keywords: ill-posed inverse problems, variational bayesian methodology, sparse signal reconstruction, infinite dimensional convex optimization