On the long-time integration of stochastic gradient systems
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...
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| Format: | Article |
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Royal Society
2014
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| Online Access: | https://eprints.nottingham.ac.uk/29287/ |
| _version_ | 1848793755221491712 |
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| author | Leimkuhler, B. Matthews, C. Tretyakov, M.V. |
| author_facet | Leimkuhler, B. Matthews, C. Tretyakov, M.V. |
| author_sort | Leimkuhler, B. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:05:20Z |
| format | Article |
| id | nottingham-29287 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:05:20Z |
| publishDate | 2014 |
| publisher | Royal Society |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-292872020-05-04T20:17:07Z https://eprints.nottingham.ac.uk/29287/ On the long-time integration of stochastic gradient systems Leimkuhler, B. Matthews, C. Tretyakov, M.V. 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. Royal Society 2014 Article PeerReviewed Leimkuhler, B., Matthews, C. and Tretyakov, M.V. (2014) On the long-time integration of stochastic gradient systems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470 (2170). 20140120/1-20140120/14. ISSN 1471-2946 stochastic gradient systems weak convergence Brownian dynamics stochastic differential equation http://rspa.royalsocietypublishing.org/content/470/2170/20140120.abstract doi:10.1098/rspa.2014.0120 doi:10.1098/rspa.2014.0120 |
| spellingShingle | stochastic gradient systems weak convergence Brownian dynamics stochastic differential equation Leimkuhler, B. Matthews, C. Tretyakov, M.V. On the long-time integration of stochastic gradient systems |
| title | On the long-time integration of stochastic gradient systems |
| title_full | On the long-time integration of stochastic gradient systems |
| title_fullStr | On the long-time integration of stochastic gradient systems |
| title_full_unstemmed | On the long-time integration of stochastic gradient systems |
| title_short | On the long-time integration of stochastic gradient systems |
| title_sort | on the long-time integration of stochastic gradient systems |
| topic | stochastic gradient systems weak convergence Brownian dynamics stochastic differential equation |
| url | https://eprints.nottingham.ac.uk/29287/ https://eprints.nottingham.ac.uk/29287/ https://eprints.nottingham.ac.uk/29287/ |