Title: Geometric MCMC for infinite-dimensional inverse problems
Authors: Alexandros Beskos - University College London (United Kingdom) [presenting]
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 finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-size 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 informed by the data. At the same time, recently developed geometric MCMC algorithms are found to be powerful in exploring complicated distributions that deviate from elliptic Gaussian laws, but are in general computationally intractable for models in infinite dimensions. We combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing computational costs. 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 arising in subsurface flow, heat conduction and incompressible flow control.