Metropolis Integration Schemes for Self-Adjoint Diffusions

Abstract

Abstract. We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order accuracy in the small noise limit. It also permits to avoid computing explicitly certain terms in the equation, such as the divergence of the mobility tensor, which can be tedious to calculate. Finally, the scheme is shown to be ergodic with respect to the exact equilibrium probability distribution of the diffusion when it exists. These results are illustrated on several examples including a Brownian dynamics simulation of DNA in a solvent. In this example, the proposed scheme is able to accurately compute dynamics at time step sizes that are an order of magnitude (or more) larger than those permitted with commonly used explicit predictor-corrector schemes.

Brownian dynamics of confined suspensions of active microrollers

“…(A2) may be useful for other Langevin equations where the system can reach an unphysical state for which the mobility is either undefined or not SPD, or both. This approach is much simpler and more efficient than trying to strictly prevent unphysical configurations, as in Metropolis schemes [60]. It is important to ensure, however, that the Heaviside regularization is only applied infrequently, and therefore does not affect the results significantly.…”

Ensemble preconditioning for Markov chain Monte Carlo simulation

“…An example implementation in Python is available at Matthews (2016). For Metropolization of overdamped schemes such as (3), see Bou-Rabee et al (2014).…”

Brownian dynamics of confined rigid bodies

“…Of course, the actual dynamics of the particles is three dimensional, and a complete theoretical or numerical analysis of the diffusive dynamics Parallel (x−x or y−y) as well as perpendicular (z−z) components of the rotational mean square displacement (34). The dashed line shows the asymptotic rotational MSD (47). We see that the rotational dynamics of the rigid icosahedron and a true sphere are also in good agreement.…”

“…This naive approach modifies the dynamics in a way that violates ergodicity and detailed balance, and we reduced our time step size to avoid performing a significant number of rejections. Several more sophisticated approaches exist that may solve this problem, including Metropolization [47], adaptive time-stepping [46], or continious-time discretizations [63]. Employing these techniques in our integrators remains an area of future exploration.…”