  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 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.
