## Bernoulli

• Bernoulli
• Volume 22, Number 1 (2016), 60-106.

### A function space HMC algorithm with second order Langevin diffusion limit

#### 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 $\mathbb{R}^{N}$; (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 $\mathbb{R}^{N}$ 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.

#### Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 60-106.

Dates
Revised: March 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620844

Digital Object Identifier
doi:10.3150/14-BEJ621

Mathematical Reviews number (MathSciNet)
MR3449777

Zentralblatt MATH identifier
1346.60119

#### 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. https://projecteuclid.org/euclid.bj/1443620844

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