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Hybrid Monte Carlo on Hilbert spaces

Beskos, A. and Pinski, F. J. and Sanz-Serna, J. M. and Stuart, A. M. (2011) Hybrid Monte Carlo on Hilbert spaces. Stochastic Processes and their Applications, 121 (10). pp. 2201-2230. ISSN 0304-4149.

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The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures π which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself. We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure Π in an enlarged phase space having the target π as a marginal, together with a Hamiltonian flow that preserves Π; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Π when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space.

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Additional Information:© 2011 Elsevier. Received 27 July 2010; received in revised form 5 May 2011; accepted 12 June 2011. Available online 24 June 2011. The work of Sanz-Serna is supported by MTM2010-18246-C03-01 (Ministerio de Ciencia e Innovacion). The work of Stuart is supported by the EPSRC and the ERC. The authors are grateful to the referees and editor for careful reading of the manuscript, and for many helpful suggestions for improvement.
Funding AgencyGrant Number
Ministerio de Ciencia e Innovacion (MINECO)MTM2010-18246-C03-01
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Subject Keywords:Hamiltonian dynamics; Splitting technique; Absolute continuity; Hybrid Monte Carlo
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Other Numbering System NameOther Numbering System ID
Andrew StuartJ87
Issue or Number:10
Record Number:CaltechAUTHORS:20160804-150437807
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Official Citation:A. Beskos, F.J. Pinski, J.M. Sanz-Serna, A.M. Stuart, Hybrid Monte Carlo on Hilbert spaces, Stochastic Processes and their Applications, Volume 121, Issue 10, October 2011, Pages 2201-2230, ISSN 0304-4149, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69451
Deposited By: Linda Taddeo
Deposited On:04 Aug 2016 23:42
Last Modified:03 Oct 2019 10:22

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