Efficient Numerical Algorithms for the Generalized Langevin Equation

Abstract

We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on canonical sampling properties of numerical integrators. For this purpose, we cast the GLE in an extended phase space formulation and derive a family of splitting methods which generalize existing Langevin dynamics integration methods. We show exponential convergence in law and the validity of a central limit theorem for the Markov chains obtained via these integration methods, and we show that the dynamics of a suggested integration scheme is consistent with asymptotic limits of the exact dynamics and can reproduce (in the short memory limit) a superconvergence property for the analogous splitting of underdamped Langevin dynamics. We then apply our proposed integration method to several model systems, including a Bayesian inference problem. We demonstrate in numerical experiments that our method outperforms other proposed GLE integration schemes in terms of the accuracy of sampling. Moreover, using a parameterization of the memory kernel in the GLE as proposed by Ceriotti et al [12], our experiments indicate that the obtained GLE-based sampling scheme outperforms state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency.

Non-Markovian systems out of equilibrium: exact results for two routes of coarse graining

“…To supplement and illustrate the above theoretical results we will also perform computer simulations of the microscopic stochastic model, equations ( 1) and (2). In recent years several ways have been suggested to discretize and integrate such kind of equations of motion [64][65][66][67]. Here, we will use an approach similar to reference [68] to derive the integrator with timestep Δt for the velocities at time v n i = v i (t = nΔt).…”

Hypoelliptic entropy dissipation for stochastic differential equations

“…Theoretically, it is one of core problems studied by probability [7,23], non-equilibrium dynamics [14,24,35,42], differential geometry [10,11] and ergodic theory [8,28,32,38]. In practice, this convergence rate is useful in studying long time behaviors of molecular dynamics [15,29,40]. Nowadays, this rate is important in estimating the speed of Markov-Chain-Monte-Carlo methods (MCMC), which is an emerging issue in AI and Bayesian sampling algorithms [15,17,13,40].…”

Non-Markovian systems out of equilibrium: Exact results for two routes of coarse graining

“…(2) and (3). In recent years several ways have been suggested to discretize and integrate such kind of equations of motion [60][61][62][63]. Here, we will use an approach similar to Ref.…”