Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

TL;DR

This paper introduces a novel randomize-then-optimize approach for sampling from Bayesian inverse problem posteriors with Besov priors, demonstrating effectiveness across various imaging tasks and analyzing parameter influences.

Contribution

It presents a new sampling method for Besov priors in Bayesian inverse problems, improving posterior exploration and mode estimation.

Findings

Effective sampling of posterior distributions with Besov priors.

Performance comparable or superior to existing methods in numerical experiments.

Insights into the influence of Besov parameters and wavelet basis choices.

Abstract

In this work, we investigate the use of Besov priors in the context of Bayesian inverse problems. The solution to Bayesian inverse problems is the posterior distribution which naturally enables us to interpret the uncertainties. Besov priors are discretization invariant and can promote sparsity in terms of wavelet coefficients. We propose the randomize-then-optimize method to draw samples from the posterior distribution with Besov priors under a general parameter setting and estimate the modes of the posterior distribution. The performance of the proposed method is studied through numerical experiments of a 1D inpainting problem, a 1D deconvolution problem, and a 2D computed tomography problem. Further, we discuss the influence of the choice of the Besov parameters and the wavelet basis in detail, and we compare the proposed method with the state-of-the-art methods. The numerical results suggest that the proposed method is an effective tool for sampling the posterior distribution equipped with general Besov priors.