Whittle-Mat\'{e}rn Gaussian random fields are popular tools in spatial statistics. Interpreting them as solutions to specific stochastic PDEs allow to generalize them from fields on all of $\mathbb{R}^d$ to bounded domains and manifolds. In this talk we focus on Riemannian manifolds and efficient approximations of Gaussian random fields based on surface finite element methods.

We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation, which allows us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class.  We demonstrate the usefulness of the results by applying them in the context of high-dimensional drift estimation and Langevin MCMC for moderately heavy-tailed target densities.

The talk presents a finite dimensional statistical model for the Calder\'{o}n problem with piecewise constant conductivities. In this setting one can consider a classical i.i.d noise model and the injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator. This results in a BvM-theorem and optimality guarantees for estimation in Bayesian posterior means.

We consider a multidimensional diffusion model describing a particle moving in an insulated inhomogeneous medium under Brownian dynamics. We study Bayesian inference based on discrete high-frequency observations of the particle’s location. Bayesian posteriors (and their posterior means) based on suitable Gaussian priors are shown to estimate the diffusivity function of the medium at the minimax optimal rate over Holder smoothness classes in any dimension. We also show that certain penalized least squares estimators are minimax optimal for estimating the diffusivity.