A function space HMC algorithm with second order Langevin diffusion limit

Abstract

We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on R N ; (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on R N in a number of steps which is independent of N. We also present the underlying theory for the limiting nonreversible diffusion on Hilbert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…The law of the solution to this SDE conditioned on the events X (0) = x − and X (s + ) = x + is a probability measure π on L 2 ([0, s + ], R n ) which poses a challenging and important sampling problem, especially if U is multimodal. This setting has been used as a test case for sampling probability measures in high dimensions (see for example [9] and [45]). For a more detailed introduction (including applications) see [11] and for a rigorous theoretical treatment the papers [11,[24][25][26].…”

Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions

“…Algorithms have also been developed that are based on second order Langevin diffusions, in which a stochastic differential equation governs the behaviour of the velocity of a process [62,63]. A natural extension to the work of Girolami and Calderhead [1] and Xifara et al [33] would be to map such diffusions onto a manifold and derive Metropolis-Hastings proposal kernels based on the resulting dynamics.…”

Reversible and non-reversible Markov chain Monte Carlo algorithms for reservoir simulation problems

“…Hamiltonian Monte Carlo (SOL-HMC) algorithm, introduced in [8]. Both of them are nonreversible modifications of the well known Hamiltonian Monte Carlo (HMC) [13], which is reversible.…”