Non-stationary multi-layered Gaussian priors for Bayesian inversion. (3rd December 2020)

• Record Type: Journal Article
• Authors: Emzir, Muhammad; Lasanen, Sari; Purisha, Zenith; Roininen, Lassi; Särkkä, Simo
• Is Part Of: Inverse problems. Volume 37:Number 1(2021)
• Publication Date: 2020-12-03
• Subjects: inverse problem -- Bayesian inverse problem -- multi-layer Gaussian field priors; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/abc962
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. The aim of the multi-layered hierarchical prior is to provide enough complexity structure to allow for both smoothing and edge-preserving properties at the same time. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We then build the computational inference method using a finite-dimensional Galerkin method. We show that the proposed approximation has a convergence-in-probability property to the solution of the original multi-layered model. We then carry out Bayesian inference using the preconditioned Crank–Nicolson algorithm which is modified to work with multi-layered Gaussian fields. We show via numerical experiments in signal deconvolution and computerized x-ray tomography problems that the proposed method can offer both smoothing and edge preservation at the same time.