Dimension-Independent MCMC algorithms for PDE-contrained BIPs

Bayesian inverse problems often involve sampling probability distributions on functions. Traditional MCMC algorithms fail under mesh refinement. Recently, a variety of dimension-independent MCMC methods have emerged, but few of them take the geometry of the posterior into account.  In this work, we try to blend recent developments in finite dimensional manifold samplers with dimension-independent MCMC. The goal of such a marriage is to speed up the mixing of the Markov chain by using geometry, whilst remaining robust under mesh-refinement. The key idea is to employ the manifold methods on an appropriately chosen finite dimensional subspace.