A Bayesian level set method for geometric inverse problems

Iglesias, Marco; Lu, Yulong; Stuart, Andrew

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 may be used to not only estimate interface locations, but quantify uncertainty in them; 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.

Citation

Iglesias, M., Lu, Y., & Stuart, A. (2016). A Bayesian level set method for geometric inverse problems. Interfaces and Free Boundaries, 18(2), https://doi.org/10.4171/IFB/362

• Journal Article Type: Article
• Acceptance Date: Mar 4, 2016
• Publication Date: May 1, 2016
• Deposit Date: Mar 1, 2017
• Publicly Available Date: Mar 1, 2017
• Journal: Interfaces and Free Boundaries
• Print ISSN: 1463-9963
• Electronic ISSN: 1463-9971
• Publisher: European Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 2
• DOI: https://doi.org/10.4171/IFB/362
• Keywords: Inverse problems, Bayesian level set method, Markov chain Monte Carlo (MCMC)
• Public URL: https://nottingham-repository.worktribe.com/output/976753
• Publisher URL: http://www.ems-ph.org/journals/show_abstract.php?issn=1463-9963&vol=18&iss=2&rank=3
• Contract Date: Mar 1, 2017