Abstract
We treat the problem of recovering the unknown shape of a single inclusion with unknown constant permittivity in an otherwise uniform background material, from uncertain measurements of trans-capacitance at electrodes outside the material. The ubiquitous presence of measurement noise implies that the practical measurement process is probabilistic, and the inverse problem is naturally stated as statistical inference. Formulating the inverse problem in a Bayesian inferential framework requires accurately modelling the forward map, measurement noise, and specifying a prior distribution for the cross-sectional material distribution. Numerical implementation of the forward map is via the boundary element method (BEM) taking advantage of a piecewise constant representation. Summary statistics are calculated using MCMC sampling to characterize posterior variability for synthetic and measured data sets.