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A function space HMC algorithm with second order Langevin diffusion limit
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Ottobre, Michela, Pillai, Natesh S., Pinski, Frank J. and Stuart, A. M. (2015) A function space HMC algorithm with second order Langevin diffusion limit. Bernoulli, 22 (1). pp. 60-106. doi:10.3150/14-BEJ621 ISSN 1350-7265.
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Official URL: http://dx.doi.org/10.3150/14-BEJ621
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 RNRN; (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 RNRN in a number of steps which is independent of NN. 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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Gaussian measures, Hilbert space, Markov processes, Monte Carlo method, Langevin equations, Algorithms | ||||||||
Journal or Publication Title: | Bernoulli | ||||||||
Publisher: | Int Statistical Institute | ||||||||
ISSN: | 1350-7265 | ||||||||
Official Date: | 30 September 2015 | ||||||||
Dates: |
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Volume: | 22 | ||||||||
Number: | 1 | ||||||||
Page Range: | pp. 60-106 | ||||||||
DOI: | 10.3150/14-BEJ621 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 3 March 2017 | ||||||||
Date of first compliant Open Access: | 3 March 2017 | ||||||||
Funder: | National Science Foundation (U.S.) (NSF), Great Britain. Office for Nuclear Regulation (ONR), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC) |
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