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A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit

Ottobre, Michela and Pillai, Natesh S. and Pinski, Frank J. and Stuart, Andrew M. (2016) A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit. Bernoulli, 22 (1). pp. 60-106. ISSN 1350-7265 . http://resolver.caltech.edu/CaltechAUTHORS:20160715-161420502

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.3150/14-BEJ621DOIArticle
http://projecteuclid.org/euclid.bj/1443620844PublisherArticle
https://arxiv.org/abs/1308.0543arXivDiscussion Paper
Additional Information:© 2016 ISI/BS. Received September 2013 and revised March 2014. NSP is partially supported by NSF and ONR grants. AMS thanks EPSRC and ERC for support.
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Subject Keywords:diffusion limits; function space Markov chain Monte Carlo; hybrid Monte Carlo algorithm; second order Langevin diffusion
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ121
Record Number:CaltechAUTHORS:20160715-161420502
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20160715-161420502
Official Citation:Ottobre, Michela; Pillai, Natesh S.; Pinski, Frank J.; Stuart, Andrew M. A function space HMC algorithm with second order Langevin diffusion limit. Bernoulli 22 (2016), no. 1, 60--106. doi:10.3150/14-BEJ621. http://projecteuclid.org/euclid.bj/1443620844.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69073
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:19 Jul 2016 00:11
Last Modified:01 Aug 2017 22:57

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