Non-Stationary Multi-layered Gaussian Priors for Bayesian Inversion

TL;DR

This paper introduces a novel Bayesian inversion approach using non-stationary multi-layered Gaussian priors, employing SPDEs and Galerkin methods, with demonstrated effectiveness in signal deconvolution and tomography.

Contribution

It develops a new computational framework for Bayesian inverse problems with multi-layered Gaussian priors, including convergence analysis and a modified inference algorithm.

Findings

Method achieves convergence-in-probability to the true model

Provides smoothing and edge preservation simultaneously

Effective in signal deconvolution and X-ray tomography

Abstract

In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We 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.