Abstract

In this talk we consider the Bayesian approach to inverse problems and infer uncertain coefficients in elliptic PDEs given noisy observations of the associated solution. After provinding a short introduction to this approach and illustrating it at a real-world groundwater flow problem, we focus on Metropolis-Hastings (MH) algorithms for approximate sampling of the resulting posterior distribution. These methods used to suffer from a high dimensional state space or a highly concentrated posterior measure, respectively.

In recent years dimension-independent MH algorithms have been developed and analyzed, suitable for Bayesian inference in infinite dimensions. However, the second issue of a concetrated posterior has drawn less attention in the study of MH algorithms yet, despite its importance in application.

We present a MH algorithm well-defined in Hilbert spaces which possesses both desirable properties: a dimension-independent performance as well as a robust behaviour w.r.t. small noise levels in the observational data. Moreover, we show a first analysis of the noise-independence of MH algorithms in terms of the expected acceptance rate and the expected squared jump distance of the resulting Markov chains. Numerical experiments confirm the theoretical results and also indicate that they hold in more general situations than proven.