Hybrid Monte Carlo methods for sampling probability measures on submanifolds

Tony Lelièvre 1, 2 Mathias Rousset 3 Gabriel Stoltz 1, 2
2 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
3 SIMSMART - SIMulation pARTiculaire de Modèles Stochastiques
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for large timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.
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https://hal.archives-ouvertes.fr/hal-01832820
Contributor : Tony Lelièvre <>
Submitted on : Monday, July 9, 2018 - 9:37:18 AM
Last modification on : Friday, April 19, 2019 - 4:55:18 PM

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  • HAL Id : hal-01832820, version 1
  • ARXIV : 1807.02356

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Tony Lelièvre, Mathias Rousset, Gabriel Stoltz. Hybrid Monte Carlo methods for sampling probability measures on submanifolds. 2018. ⟨hal-01832820⟩

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