Volume 38, Issue 3
Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations

J. Comp. Math., 38 (2020), pp. 487-501.

Published online: 2020-03

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• Abstract

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.

• Keywords

Stochastic differential equations, Parareal algorithm, Convergence, Stochastic Taylor expansion, Milstein scheme.

60H10, 60H35, 65Y05

lyzhang@lsec.cc.ac.cn (Liying Zhang)

1014944214@qq.com (Jing Wang)

weienzhou@nudt.edu.cn (Weien Zhou)

Liuld@cumtb.edu.cn (Landong Liu)

lytzhangli@buu.edu.cn (Li Zhang)

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@Article{JCM-38-487, author = {Liying and Zhang and lyzhang@lsec.cc.ac.cn and 6725 and School of Mathematical Science, China University of Mining and Technology, Beijing, China and Liying Zhang and Jing and Wang and 1014944214@qq.com and 10844 and Department of Chemistry, Liaoning University, Shenyang 110036, P. R. China and Jing Wang and Weien and Zhou and weienzhou@nudt.edu.cn and 6726 and National Innovation Institute of Defense Technology, Chinese Academy of Military Science, Beijing 100101, China and Weien Zhou and Landong and Liu and Liuld@cumtb.edu.cn and 7236 and School of Mathematical Science, China University of Mining and Technology, Beijing 100083, China and Landong Liu and Li and Zhang and lytzhangli@buu.edu.cn and 7237 and Department of Foundation Courses, Beijing Union University, Beijing 100101, China and Li Zhang}, title = {Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {3}, pages = {487--501}, abstract = {

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1901-m2018-0085}, url = {http://global-sci.org/intro/article_detail/jcm/15797.html} }
TY - JOUR T1 - Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations AU - Zhang , Liying AU - Wang , Jing AU - Zhou , Weien AU - Liu , Landong AU - Zhang , Li JO - Journal of Computational Mathematics VL - 3 SP - 487 EP - 501 PY - 2020 DA - 2020/03 SN - 38 DO - http://doi.org/10.4208/jcm.1901-m2018-0085 UR - https://global-sci.org/intro/article_detail/jcm/15797.html KW - Stochastic differential equations, Parareal algorithm, Convergence, Stochastic Taylor expansion, Milstein scheme. AB -

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.

Liying Zhang, Jing Wang, Weien Zhou, Landong Liu & Li Zhang. (2020). Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations. Journal of Computational Mathematics. 38 (3). 487-501. doi:10.4208/jcm.1901-m2018-0085
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