Hierarchical Bayesian Level Set Inversion

TL;DR

This paper introduces a hierarchical Bayesian level set method for interface inversion that overcomes scale sensitivity issues, leading to more robust and efficient uncertainty quantification in inverse problems.

Contribution

It develops a hierarchical approach with a scalar parameter to improve Bayesian level set inversion, along with Gibbs-based MCMC algorithms for better performance.

Findings

Hierarchical approach reduces scale sensitivity in Bayesian inversion.

Gibbs-based MCMC methods outperform non-hierarchical counterparts.

Method effectively quantifies uncertainty in interface reconstruction.

Abstract

The level set approach has proven widely successful in the study of inverse problems for interfaces, since its systematic development in the 1990s. Recently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be circumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful consideration of the development of algorithms which encode probability measure equivalences as the hierarchical parameter is varied, this leads to well-defined Gibbs based MCMC methods found by alternating Metropolis-Hastings updates of the level set function and the hierarchical parameter. These methods demonstrably outperform non-hierarchical Bayesian level set methods.