A Bayesian approach for consistent reconstruction of inclusions

TL;DR

This paper introduces a Bayesian method for reliably reconstructing inclusions in inverse problems, demonstrating convergence of the posterior distribution and effective detection in photoacoustic tomography.

Contribution

It develops a general Bayesian framework using push-forward priors for inclusion detection, proving posterior convergence and applicability to various inverse problems.

Findings

Posterior mean converges to the true inclusion in a probabilistic sense.

Method effectively detects inclusions with regular boundaries.

Numerical tests confirm convergence and detection capabilities.

Abstract

This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any H\"older continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under H\"older continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.