  Bayesian inverse problems often involve sampling posterior distributions on
infinite-dimensional function spaces. Traditional Markov chain Monte Carlo
(MCMC) algorithms are characterized by deteriorating mixing times upon
mesh-refinement, when the finite-dimensional approximations become more
accurate. Such methods are typically forced to reduce step-sizes as the
discretization gets finer, and 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 have been found to be powerful in exploring complicated
distributions that deviate significantly from elliptic Gaussian laws, but are
in general computationally intractable for models defined in infinite
dimensions. In this work, 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 the
computational cost per step (for instance, in comparison with the vanilla
preconditioned Crank-Nicolson (pCN) method). 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 three challenging inverse problems arising in subsurface flow, heat
conduction and incompressible flow control. The algorithms exhibit up to two
orders of magnitude improvement in sampling efficiency when compared with the
pCN method.
