Abstract
In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures.
Highlights
Introduction and MotivationSampling from probability measures in high-dimensional spaces is a problem that appears frequently in applications, e.g. in computational statistical mechanics and in Bayesian statistics
As is typical in many applications, in molecular dynamics and Bayesian inference, the density is known only up to a
A standard approach to approximating such integrals is Markov Chain Monte Carlo (MCMC) techniques [19,32,52], where a Markov process (Xt )t≥0 is constructed which is ergodic with respect to the probability measure π
Summary
Sampling from probability measures in high-dimensional spaces is a problem that appears frequently in applications, e.g. in computational statistical mechanics and in Bayesian statistics. Our objective is to investigate the use of these dynamics for computing ergodic averages of the form (2) To this end, we study the long time behaviour of (8) and, using hypocoercivity techniques, prove that the process converges exponentially fast to equilibrium. We study the long time behaviour of (8) and, using hypocoercivity techniques, prove that the process converges exponentially fast to equilibrium This perturbed underdamped Langevin process introduces a number of parameters in addition to the mass and friction tensors which must be tuned to ensure that the process is an efficient sampler.
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