I will begin by overviewing the Bayesian approach
to the reconstruction of fields from indirect
and noisy (possibly nonlinear) measurement functionals .
I will then explain the basic Bayesian level set approach to reconstructing piecewise constant fields .
Finally I will demonstrate how the method can be enhanced
by means of a hierarchical multiscale approach in which the
charateristic length scale of interface separation is
learned from the data, along with the geometry of interfaces themselves.
 M. Dashti and A.M. Stuart. The Bayesian approach to inverse problems.
To appear in Handbook of Uncertainty Quantification, Springer, 2016.
 M.A. Iglesias, Y. Lu, A.M. Stuart, "A level-set approach to Bayesian
geometric inverse problems", submitted.
Joint work with Matt Dunlop (Warwick) and Marco Iglesias (Nottingham)