Strong convergence rates of probabilistic integrators for ordinary differential equations
- 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. ), 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-squareProbabilistic 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. ), 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.…
Author details: | Han Cheng LieORCiD, A. M. Stuart, Tim J. SullivanORCiDGND |
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DOI: | https://doi.org/10.1007/s11222-019-09898-6 |
ISSN: | 0960-3174 |
ISSN: | 1573-1375 |
Title of parent work (English): | Statistics and Computing |
Publisher: | Springer |
Place of publishing: | Dordrecht |
Publication type: | Article |
Language: | English |
Date of first publication: | 2019/10/22 |
Publication year: | 2019 |
Release date: | 2020/10/20 |
Tag: | Convergence rates; Ordinary differential equations; Probabilistic numerical methods; Uncertainty quantification |
Volume: | 29 |
Issue: | 6 |
Number of pages: | 19 |
First page: | 1265 |
Last Page: | 1283 |
Funding institution: | Freie Universitat Berlin within the Excellence Initiative of the German Research FoundationGerman Research Foundation (DFG); Universitat Potsdam; DARPAUnited States Department of DefenseDefense Advanced Research Projects Agency (DARPA); EPSRCEngineering & Physical Sciences Research Council (EPSRC); ONROffice of Naval Research; National Science FoundationNational Science Foundation (NSF) [DMS-1127914] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access |
Open Access / Green Open-Access |