Abstract
Many Bayesian inference problems require exploring the posterior distribution of highdimensional
parameters that represent the discretization of an underlying function. This
work introduces a family of Markov chain Monte Carlo (MCMC) samplers that can adapt
to the particular structure of a posterior distribution over functions. Two distinct lines
of research intersect in the methods developed here. First, we introduce a general class
of operator-weighted proposal distributions that are well defined on function space, such
that the performance of the resulting MCMC samplers is independent of the discretization
of the function. Second, by exploiting local Hessian information and any associated lowdimensional
structure in the change from prior to posterior distributions, we develop an
inhomogeneous discretization scheme for the Langevin stochastic differential equation that
yields operator-weighted proposals adapted to the non-Gaussian structure of the posterior.
The resulting dimension-independent and likelihood-informed (DILI) MCMC samplers may
be useful for a large class of high-dimensional problems where the target probability
measure has a density with respect to a Gaussian reference measure. Two nonlinear inverse
problems are used to demonstrate the efficiency of these DILI samplers: an elliptic PDE
coefficient inverse problem and path reconstruction in a conditioned diffusion.