A Bayesian Level Set Method for Geometric Inverse Problems

TL;DR

This paper presents a Bayesian level set approach for geometric inverse problems, enabling well-posedness and efficient computation of interfaces in applications like medical imaging and subsurface modeling.

Contribution

It introduces a Bayesian formulation for level set methods that ensures well-posedness and allows implicit interface updates via MCMC without explicit velocity fields.

Findings

Posterior distribution is Lipschitz continuous with respect to data.

Develops computational algorithms using MCMC for level set updates.

Demonstrates applications in subsurface modeling and inverse source problems.

Abstract

We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC methodology applied to the level set function- no explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications.