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A Bayesian level set method for geometric inverse problems. (English) Zbl 1353.65050

Summary: 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 Markov chain Monte Carlo 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.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65J22 Numerical solution to inverse problems in abstract spaces
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
92C55 Biomedical imaging and signal processing
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References:

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