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On the long-time integration of stochastic gradient systems

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

Authors

B. Leimkuhler

C. Matthews

M.V. Tretyakov



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.

Journal Article Type Article
Publication Date Jan 1, 2014
Journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Print ISSN 1364-5021
Electronic ISSN 1471-2946
Publisher Royal Society, The
Peer Reviewed Peer Reviewed
Volume 470
Issue 2170
Article Number 20140120
APA6 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), doi:10.1098/rspa.2014.0120
DOI https://doi.org/10.1098/rspa.2014.0120
Keywords stochastic gradient systems, weak convergence, Brownian dynamics, stochastic differential equation
Publisher URL http://rspa.royalsocietypublishing.org/content/470/2170/20140120.abstract
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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