2025/06/23 by Andreas Horst, Babak Maboudi Afkham, Babak M. Afkham +6 · 1 voice
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Bayesian probability #Besov space #Deconvolution #Gaussian #Gibbs sampling #Inverse problem #Piecewise #Prior probability #Smoothing #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #math.NA #stat.CO
paper · pdf · open access · doi:10.1016/j.apnum.2026.05.015
published in Applied Numerical Mathematics 229, 87-106 (Elsevier BV)
arxiv published 2025/06/23 · openalex publication_date 2026/06/01 · openalex created_date 2026/06/02 · arxiv updated 2026/07/24 · openalex updated_date 2026/08/05
In many inverse problems, the unknown is composed of multiple components with different regularities, for example, in imaging problems, where the unknown can have both rough and smooth features. We investigate linear Bayesian inverse problems, where the unknown consists of two components: one smooth and one piecewise constant. We model the unknown as a sum of two components and assign individual priors on each component to impose the assumed behavior. We propose and compare two prior models: (i) a combination of a Haar wavelet-based Besov prior and a smoothing Besov prior, and (ii) a hierarchical Gaussian prior on the gradient coupled with a smoothing Besov prior. To achieve a balanced reconstruction, we place hyperpriors on the prior parameters and jointly infer both the components and the hyperparameters. We propose Gibbs sampling schemes for posterior inference in both prior models. We demonstrate the capabilities of our approach on 1D and 2D deconvolution problems, where the unknown consists of smooth parts with jumps. The numerical results indicate that our methods improve the reconstruction quality compared to single-prior approaches and that the prior parameters can be successfully estimated to yield a balanced decomposition.