A Bayesian level set method for an inverse medium scattering problem in acoustics

TL;DR

This paper introduces a Bayesian level set approach for determining the shape of scatterers in 2D acoustic inverse medium problems, utilizing MCMC sampling to estimate the posterior distribution of shapes.

Contribution

It develops a Bayesian framework with level set functions for shape reconstruction in acoustic scattering, including analysis of posterior well-posedness and numerical validation.

Findings

Effective shape reconstruction demonstrated through numerical experiments

Bayesian approach provides probabilistic shape estimates

MCMC sampling successfully explores the posterior distribution

Abstract

In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities, and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution would be discussed, and the Markov chain Monte Carlo (MCMC) methods will be applied to generate samples from the arising posterior distribution. Numerical experiments will be presented to demonstrate the effectiveness of the proposed method.