Dimension-independent likelihood-informed MCMC

TL;DR

This paper introduces dimension-independent, likelihood-informed MCMC methods that adapt to the structure of high-dimensional function space posteriors, improving sampling efficiency in complex Bayesian inverse problems.

Contribution

It develops a new class of operator-weighted proposals and an inhomogeneous Langevin scheme that are independent of discretization and exploit local structure for better sampling.

Findings

Efficient sampling demonstrated on elliptic PDE inverse problem.

Effective path reconstruction in conditioned diffusion.

Performance remains stable across different discretizations.

Abstract

Many Bayesian inference problems require exploring the posterior distribution of high-dimensional 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 low-dimensional 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, 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.