Title: Geometric MCMC for infinite-dimensional inverse problems
Authors: Alexandros Beskos - University College London (United Kingdom) [presenting]
Andrew Stuart - California Institute of Technology (United States)
Mark Girolami - University of Cambridge (United Kingdom)
Patrick Farell - University of Oxford (United Kingdom)
Shiwei Lan - University of Illinois Urbana-Champaign (United States)
Abstract: Bayesian Inverse Problems often involve sampling posteriors on infinite-dimensional spaces. Traditional MCMC algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are forced to reduce step-sizes as the discretization gets finer, thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior. At the same time, geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate from elliptic Gaussian laws, but are computationally intractable for models defined in infinite-dimensions. We combine geometric methods on finite-dimensional subspaces with mesh-independent infinite-dimensional approaches. The objective is to speed up MCMC mixing, without significantly increasing the computational cost per-step. This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of 3 challenging Inverse Problems and can exhibit up to two orders of magnitude improvement in sampling efficiency when compared with pCN.