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Strong convergence rates of probabilistic integrators for ordinary differential equations

Lie, Han Cheng and 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. ISSN 0960-3174.

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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-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.

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Additional Information:© 2019 Springer Science+Business Media, LLC, part of Springer Nature. First Online: 22 October 2019. HCL and TJS are supported by the Freie Universität Berlin within the Excellence Initiative of the German Research Foundation. HCL is supported by the Universität Potsdam. AMS is grateful to DARPA, EPSRC and ONR for funding. This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of these funding agencies and institutions.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
Universität PotsdamUNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Subject Keywords:Probabilistic numerical methods; Ordinary differential equations; Convergence rates; Uncertainty quantification
Issue or Number:6
Classification Code:2010 Mathematics Subject Classification: 65L20, 65C99, 37H10, 68W20
Record Number:CaltechAUTHORS:20170612-123841285
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Official Citation:Lie, H.C., Stuart, A.M. & Sullivan, T.J. Stat Comput (2019) 29: 1265.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78107
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:58
Last Modified:04 Nov 2019 22:37

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