- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:04
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
2
A Bayesian level set method for geometric inverse problems
Marco
Iglesias
University of Nottingham, NOTTINGHAM, UNITED KINGDOM
Yulong
Lu
University of Warwick, COVENTRY, UNITED KINGDOM
Andrew
Stuart
University of Warwick, COVENTRY, UNITED KINGDOM
Inverse problems, Bayesian level set method, Markov chain Monte Carlo (MCMC)
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.
Integral equations
Partial differential equations
Statistics
181
217
10.4171/IFB/362
http://www.ems-ph.org/doi/10.4171/IFB/362