Efficient numerical algorithms for the generalized Langevin equation

We study the design and implementation of numerical methods to solve the generalized Langevin equation (GLE) focusing on the sampling properties of the numerical integrators. For this purpose, we cast the GLE in an extended phase space formulation and derive a family of splitting methods that 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, we show that 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, Bussi, and Parrinello Phys. Rev. Lett., 6 (2010), pp. 1170–1180, our experiments indicate that the obtained GLE-based sampling scheme can, in some cases, outperform state-of-the-art sampling schemes based on underdamped Langevin dynamics in terms of robustness and efficiency.