A variational Bayesian method for inverse problems with impulsive noise

TL;DR

This paper introduces a Bayesian variational method using a heavy-tailed t distribution to robustly solve inverse problems affected by impulsive noise and outliers, with automatic hyper-parameter determination and demonstrated effectiveness on heat conduction problems.

Contribution

It presents a novel variational Bayesian approach employing a heavy-tailed t distribution for robust inverse problem solving with automatic hyper-parameter estimation.

Findings

Robustness to outliers demonstrated in numerical experiments.

Fast and steady convergence of the proposed algorithm.

Effective in both linear and nonlinear inverse heat conduction problems.

Abstract

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm.