Solving linear Bayesian inverse problems using a fractional total variation-Gaussian (FTG) prior and transport map

TL;DR

This paper introduces a fractional total variation-Gaussian prior for linear Bayesian inverse problems involving functions with sharp features, and develops an efficient transport map-based sampler for high-dimensional posterior sampling.

Contribution

It proposes a novel FTG prior combining fractional total variation with Gaussian measures and a transport map-based sampler for improved Bayesian inference in function space.

Findings

FTG prior effectively captures detailed features without staircase effects.

Transport map sampler's acceptance probability is independent of discretization dimension.

Numerical examples show the method's robustness and efficiency.

Abstract

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. In particular, when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.