Statistical Estimation of the Parameters of a PDE

We consider the class of image recovery problems where the desired image appears as the spatially-varying coefficients of a partial differential equation (PDE), the data consists of measured values on the boundary, and the forward map can only be adequately simulated by solving the PDE subject to boundary conditions. This is the natural mathematical setting for non-invasive imaging using strongly scattered waves. We give a gentle introduction to the statistical (Bayesian) approach to solving this class of inverse problems, and present some computational examples using sampling algorithms. The statistical approach quantifies the inherent uncertainty in images recovered from incomplete noisey data using knowledge of the forward map the measurement process and noise statistics. The single most likely image, usually found by applying the inverse of the forward map to the measured data, does not give a good reconstruction in this class of problems because that image is unrepresentative of the bulk of feasible images. Instead it is necessary to summarize the feasible images by calculating expectations over the posterior distribution. One advantage of this route to solving inverse problems is the ability to quantify accuracy within the recovered image, while a major disadvantage is computational expense. We use the illustrative example of imaging electrical conductivity and give examples of reconstruction of an unknown conductivity from simple synthetic data. In principle, high-level models may be incorporated relatively easily in the sampling algorithms, and we give a glimpse of some image models that are currently used.