A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations

TL;DR

This paper introduces a TV-Gaussian prior for infinite-dimensional Bayesian inverse problems that effectively models functions with sharp jumps, and proposes an efficient MCMC sampling method, demonstrating its advantages through numerical examples.

Contribution

The paper develops a novel TV-Gaussian prior suitable for infinite-dimensional spaces and provides an efficient MCMC algorithm for sampling from the resulting posterior.

Findings

The TG prior effectively captures sharp jumps in functions.

The proposed MCMC algorithm shows high efficiency in sampling.

Numerical examples validate the method's performance.

Abstract

Many scientific and engineering problems require to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, choosing an appropriate prior distribution is an important task. In particular we consider problems where the function to infer is subject to sharp jumps which render the commonly used Gaussian measures unsuitable. On the other hand, the so-called total variation (TV) prior can only be defined in a finite dimensional setting, and does not lead to a well-defined posterior measure in function spaces. In this work we present a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect sharp jumps of the function, and the Gaussian distribution is used as a reference measure so that it results in a well-defined posterior measure in the function space. We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the TG prior. With numerical examples we demonstrate the performance of the TG prior and the efficiency of the proposed MCMC algorithm.