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.
Classification :
45-XX, 35-XX, 62-XX
Mots-clés :
Inverse problems, Bayesian level set method, Markov chain Monte Carlo (MCMC)
Affiliations des auteurs :
Marco A. Iglesias
1
;
Yulong Lu
2
;
Andrew M. Stuart
3
1
University of Nottingham, UK
2
University of Warwick, Coventry, UK
3
University of Warwick, Coventry, United Kingdom
Marco A. Iglesias; Yulong Lu; Andrew M. Stuart. A Bayesian level set method for geometric inverse problems. Interfaces and free boundaries, Tome 18 (2016) no. 2, pp. 181-217. doi: 10.4171/ifb/362
@article{10_4171_ifb_362,
author = {Marco A. Iglesias and Yulong Lu and Andrew M. Stuart},
title = {A {Bayesian} level set method for geometric inverse problems},
journal = {Interfaces and free boundaries},
pages = {181--217},
year = {2016},
volume = {18},
number = {2},
doi = {10.4171/ifb/362},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/362/}
}
TY - JOUR
AU - Marco A. Iglesias
AU - Yulong Lu
AU - Andrew M. Stuart
TI - A Bayesian level set method for geometric inverse problems
JO - Interfaces and free boundaries
PY - 2016
SP - 181
EP - 217
VL - 18
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/362/
DO - 10.4171/ifb/362
ID - 10_4171_ifb_362
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%V 18
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%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/362/
%R 10.4171/ifb/362
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