Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions

Leimkuhler, Benedict; Sharma, Akash; Tretyakov, Michael V.

Abstract

A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, that is, expectations with respect to the invariant law of RSDEs, both inside a domain in Rd and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.

Citation

Leimkuhler, B., Sharma, A., & Tretyakov, M. V. (2023). Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions. Annals of Applied Probability, 33(3), 1904-1960. https://doi.org/10.1214/22-AAP1856

• Journal Article Type: Article
• Acceptance Date: Jun 9, 2022
• Online Publication Date: May 2, 2023
• Publication Date: Jun 1, 2023
• Deposit Date: Jun 15, 2022
• Publicly Available Date: May 2, 2023
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Electronic ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics (IMS)
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 3
• Pages: 1904-1960
• DOI: https://doi.org/10.1214/22-AAP1856
• Keywords: Ergodic limits, Neumann boundary value problem, reflected Brownian dynamics, reflected stochastic differential equations, sampling from distributions with compact support, sampling on manifold, stochastic gradient system in bounded domains, weak approxima
• Public URL: https://nottingham-repository.worktribe.com/output/8496983
• Publisher URL: https://projecteuclid.org/journals/annals-of-applied-probability/volume-33/issue-3/Simplest-random-walk-for-approximating-Robin-boundary-value-problems-and/10.1214/22-AAP1856.short