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

A function space HMC algorithm with second order Langevin diffusion limit

Michela Ottobre, Natesh S. Pillai, Frank J. Pinski, and Andrew M. Stuart

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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.

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

Received: September 2013
Revised: March 2014
First available in Project Euclid: 30 September 2015

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Zentralblatt MATH identifier

diffusion limits function space Markov chain Monte Carlo hybrid Monte Carlo algorithm second order Langevin diffusion


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.

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  • [1] Berger, E. (1986). Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Theory Relat. Fields 71 517–552.
  • [2] Beskos, A., Pinski, F.J., Sanz-Serna, J.M. and Stuart, A.M. (2011). Hybrid Monte Carlo on Hilbert spaces. Stochastic Process. Appl. 121 2201–2230.
  • [3] Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008). MCMC methods for diffusion bridges. Stoch. Dyn. 8 319–350.
  • [4] Bou-Rabee, N. and Vanden-Eijnden, E. (2010). Pathwise accuracy and ergodicity of metropolized integrators for SDEs. Comm. Pure Appl. Math. 63 655–696.
  • [5] Bou-Rabee, N. and Vanden-Eijnden, E. (2012). A patch that imparts unconditional stability to explicit integrators for Langevin-like equations. J. Comput. Phys. 231 2565–2580.
  • [6] Cotter, S.L., Roberts, G.O., Stuart, A.M. and White, D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster. Statist. Sci. 28 424–446.
  • [7] Diaconis, P., Holmes, S. and Neal, R.M. (2000). Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10 726–752.
  • [8] Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D. (1987). Hybrid Monte Carlo. Physics Letters B 195 216–222.
  • [9] Hairer, M., Stuart, A. and Voss, J. (2011). Signal processing problems on function space: Bayesian formulation, stochastic PDEs and effective MCMC methods. In The Oxford Handbook of Nonlinear Filtering (D. Crisan and B. Rozovsky, eds.). 833–873. Oxford Univ. Press, Oxford.
  • [10] Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • [11] Horowitz, A.M. (1991). A generalized guided Monte Carlo algorithm. Physics Letters B 268 247–252.
  • [12] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S.J. (1993). Accelerating Gaussian diffusions. Ann. Appl. Probab. 3 897–913.
  • [13] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S.-J. (2005). Accelerating diffusions. Ann. Appl. Probab. 15 1433–1444.
  • [14] Mattingly, J.C., Pillai, N.S. and Stuart, A.M. (2012). Diffusion limits of the random walk Metropolis algorithm in high dimensions. Ann. Appl. Probab. 22 881–930.
  • [15] Neal, R.M. (2011). MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC Handb. Mod. Stat. Methods 113–162. CRC Press, Boca Raton, FL.
  • [16] Otto, F., Weber, H. and Westdickenberg, M. (2013). Invariant measure of the stochastic Allen–Cahn equation: The regime of small noise and large system size. Available at arXiv:1301.0408.
  • [17] Pillai, N.S., Stuart, A.M. and Thiery, A.H. (2014). Noisy gradient flow from a random walk in Hilbert space. Stoch. Partial Difer. Equ. Anal. Comput. 2 196–232.
  • [18] Pillai, N.S., Stuart, A.M. and Thiéry, A.H. (2012). Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions. Ann. Appl. Probab. 22 2320–2356.
  • [19] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Berlin: Springer.
  • [20] Sanz-Serna, J.M. and Calvo, M.P. (1994). Numerical Hamiltonian Problems. Applied Mathematics and Mathematical Computation 7. London: Chapman & Hall.
  • [21] Stuart, A.M. (2010). Inverse problems: A Bayesian perspective. Acta Numer. 19 451–559.