Efficient Numerical Algorithms for the Generalized Langevin Equation

TL;DR

This paper introduces a family of splitting methods for numerically solving the generalized Langevin equation, demonstrating improved sampling accuracy and efficiency over existing schemes through theoretical analysis and numerical experiments.

Contribution

It develops a new class of splitting algorithms for the GLE, proving their convergence properties and superior sampling performance compared to prior methods.

Findings

Exponential convergence in law of the proposed integrators

The new methods outperform existing GLE schemes in sampling accuracy

Enhanced robustness and efficiency in Bayesian inference applications

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 [9], 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.