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Hybrid Monte Carlo on Hilbert spaces
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Beskos, Alexandros, Pinski, F. J., SanzSerna, J. M. and Stuart, A. M.. (2011) Hybrid Monte Carlo on Hilbert spaces. Stochastic Processes and their Applications, Vol.121 (No.10). pp. 22012230. ISSN 03044149

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Official URL: http://dx.doi.org/10.1016/j.spa.2011.06.003
Abstract
The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, highdimensional 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 finitedimensional approximations of measures pi which have density with respect to a Gaussian measure on an infinitedimensional 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 infinitedimensional setting of a Hilbert space: (i) construction of a probability measure Pi in an enlarged phase space having the target pi as a marginal, together with a Hamiltonian flow that preserves Pi; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Pi 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.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Monte Carlo method, Hilbert space  
Journal or Publication Title:  Stochastic Processes and their Applications  
Publisher:  Elsevier BV * NorthHolland  
ISSN:  03044149  
Official Date:  2011  
Dates: 


Volume:  Vol.121  
Number:  No.10  
Page Range:  pp. 22012230  
Identifier:  10.1016/j.spa.2011.06.003  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Funder:  Spain. Ministerio de Ciencia e Innovación (MICINN), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC)  
Grant number:  MTM201018246C0301 (MICINN)  
References:  [1] A. Beskos, N.S. Pillai, G.O. Roberts, J.M. SanzSerna, A.M. Stuart, optimal tuning of hybrid MonteCarlo, 

URI:  http://wrap.warwick.ac.uk/id/eprint/38357 
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