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Statistics and Computing

, Volume 29, Issue 6, pp 1265–1283 | Cite as

Strong convergence rates of probabilistic integrators for ordinary differential equations

  • Han Cheng LieEmail author
  • A. M. Stuart
  • T. J. Sullivan
Article

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.

Keywords

Probabilistic numerical methods Ordinary differential equations Convergence rates Uncertainty quantification 

Mathematics Subject Classification

65L20 65C99 37H10 68W20 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversität PotsdamPotsdam OT GolmGermany
  2. 2.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  3. 3.Institute of Mathematics, Freie Universität BerlinBerlinGermany
  4. 4.Zuse Institute BerlinBerlinGermany

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