In models involving a Gaussian field one frequently assumes the covariance
operator to be given by a negative fractional power of a second-order
elliptic differential operator of the form L:= -∇·(A∇) + κ².
Whittle-Matérn fields form an well-known example of such a model. Such
covariance operators allow for a reasonable amount of model flexibility
(adjustable correlation length and the smoothness of the field) whilst
being relatively easy to simulate. Most importantly, they allow for the
simulation of non-stationary random fields.
In our work we established optimal strong convergence rates in Hölder and Sobolov norms for Galerkin approximations of such Gaussian random fields. More specifically, we considered both spectral Galerkin methods and finite element methods. The latter, although significantly more tedious to analyse, are more suitable for non-stationary fields on non-standard domains. The talk concerns joint work with Kristin Kirchner (ETHZ).