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Strong convergence rates of probabilistic integrators for ordinary differential equations
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Lie, Han Cheng, Stuart, A. M. and Sullivan, T. J. (2019) Strong convergence rates of probabilistic integrators for ordinary differential equations. Statistics and Computing, 29 (6). pp. 1265-1283. doi:10.1007/s11222-019-09898-6 ISSN 0960-3174.
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Official URL: http://dx.doi.org/10.1007/s11222-019-09898-6
Abstract
Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat Comput 27(4):1065–1082, 2017. https://doi.org/10.1007/s11222-016-9671-0), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) |
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| Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics |
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| Library of Congress Subject Headings (LCSH): | Probabilities , Differential equations , Convergence, Inequalities (Mathematics) | |||||||||||||||
| Journal or Publication Title: | Statistics and Computing | |||||||||||||||
| Publisher: | Springer | |||||||||||||||
| ISSN: | 0960-3174 | |||||||||||||||
| Official Date: | November 2019 | |||||||||||||||
| Dates: |
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| Volume: | 29 | |||||||||||||||
| Number: | 6 | |||||||||||||||
| Page Range: | pp. 1265-1283 | |||||||||||||||
| DOI: | 10.1007/s11222-019-09898-6 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Statistics and Computing. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11222-019-09898-6 | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
| Date of first compliant deposit: | 15 April 2020 | |||||||||||||||
| Date of first compliant Open Access: | 22 October 2020 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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