Efficient numerical algorithms for the generalized Langevin equation

Benedict Leimkuhler, Matthias Sachs

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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.

Original languageEnglish
Pages (from-to)A364-A388
JournalSIAM Journal on Scientific Computing
Issue number1
Publication statusPublished - 14 Feb 2022

Bibliographical note

Funding Information:
∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section December 17, 2020; accepted for publication (in revised form) August 30, 2021; published electronically February 14, 2022. https://doi.org/10.1137/20M138497X Funding: This work was supported by the European Research Council grant 320823. The first author was further supported by the Engineering and Physical Sciences Research Council under grant EPSRC EP/P006175/1, “Data-Driven Coarse-Graining using Space-Time Diffusion Maps.” The work of the second author was supported by the Statistical and Applied Mathematical Sciences Institute under grant DMS-1638521 and by Duke University.

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics


  • Markov chain Monte Carlo
  • generalized Langevin dynamics
  • symmetric splitting methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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