Partial differential equations (PDEs) govern many natural phenomena. When trying to understand the parameters driving these phenomena, we must be aware of the inevitable errors in our measurements; in statistical inverse problems these measurement errors are modelled by statistical noise. One approach to recovering the PDE coefficients governing such statistical inverse problems is through Bayesian methodology. This thesis investigates the theoretical performance of the Bayesian approach in two particular cases.

The first model considered is the advection-diffusion equation. Kolmogorov’s equations link this partial differential equation to a corresponding (time-homogeneous) stochastic differential equation, in which a diffusion process flows according to a ‘drift function’ and is buffeted by a Brownian motion effect of spatially varying magnitude; this diffusion formulation forms the focus herein. Assuming the diffusion coefficient (the magnitude of the Brownian effect) is given, this thesis considers the problem of recovering the drift function from observations of the diffusion at discrete time intervals.

Chapter 2 gives explicit conditions on priors under which the corresponding Bayesian posteriors provably contract in $L2$ distance, as data is collected, around the true drift function, at the frequentist minimax rate (up to logarithmic factors) over periodic Besov smoothness classes. These conditions are verified for some natural nonparametric priors, some of which are shown to adapt to an unknown smoothness parameter. The results are given in the high-frequency regime, where the diffusion is observed to a later time horizon and at ever closer intervals, but in fact the minimax rate (again up to logarithmic factors) is also attained in the low-frequency regime, where the intervals between samples remain fixed. This yields the first drift estimator robust to the sampling regime.

The second model considered is the Calderón problem. This is the mathematical formulation of electrical impedance tomography, in which electrodes are attached to a patient’s skin and used to apply voltages and record the corresponding current fluxes. The current flux corresponds to the Neumann data for the solution to a PDE, governed by an interior ‘conductivity parameter’, in which the voltage gives the Dirichlet boundary values. Varying the applied voltage, we consider observing the ‘Dirichlet-to-Neumann map’, and attempt to recover the interior conductivity. The data considered in Chapter 3 consists of the Dirichlet-to-Neumann map corrupted by additive Gaussian noise. A prior is exhibited for which the posterior mean statistically converges to the true conductivity (as the noise level is taken to 0) at a near-optimal rate.

The introductory chapter outlines the minimax framework by which the posteriors are judged, and provides the background material relevant to this thesis. Of particular interest may be the included proof, in an general inverse problem setting, of natural conditions under which the consistency of the posterior mean can be guaranteed.