On the long-time integration of stochastic gradient systems

Leimkuhler, B.; Matthews, C.; Tretyakov, M.V.

Abstract

This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h → 0) convergence behavior of the error of finite time averages. Recently it has been demonstrated, by study of Fokker-Planck operators, that a non-Markovian numerical method [Leimkuhler and Matthews, 2013] generates approximations in the long time limit with higher accuracy order (2nd order) than would be expected from its weak convergence analysis (finite-time averages are 1st order accurate). In this article we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to 2nd order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler-Maruyama method, the popular 2nd order Heun method, and the non-Markovian method.

Citation

Leimkuhler, B., Matthews, C., & Tretyakov, M. (2014). On the long-time integration of stochastic gradient systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470(2170), Article 20140120. https://doi.org/10.1098/rspa.2014.0120

• Journal Article Type: Article
• Publication Date: Jan 1, 2014
• Deposit Date: Jul 7, 2015
• Publicly Available Date: Jul 7, 2015
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Electronic ISSN: 1471-2946
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 470
• Issue: 2170
• Article Number: 20140120
• DOI: https://doi.org/10.1098/rspa.2014.0120
• Keywords: stochastic gradient systems, weak convergence, Brownian dynamics, stochastic differential equation
• Public URL: https://nottingham-repository.worktribe.com/output/999099
• Publisher URL: http://rspa.royalsocietypublishing.org/content/470/2170/20140120.abstract