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Strong convergence rates of probabilistic integrators for ordinary differential equations

Lie, H. C. and Stuart, A. M. and Sullivan, T. J. (2017) Strong convergence rates of probabilistic integrators for ordinary differential equations. . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20170612-123841285

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Abstract

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced 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., 2016), 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.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/1703.03680arXivDiscussion Paper
Additional Information:HCL and TJS are supported by the Free University of Berlin within the Excellence Initiative of the German Research Foundation (DFG). AMS is grateful to DARPA, EPSRC and ONR for funding.
Funders:
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)UNSPECIFIED
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, dissipative systems, Burkholder–Davis–Gundy inequalities, uncertainty quantification
Classification Code:2010 Mathematics Subject Classification: 65L20, 65C99, 37H10, 68W20
Record Number:CaltechAUTHORS:20170612-123841285
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170612-123841285
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78107
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2017 20:58
Last Modified:12 Jun 2017 20:58

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