@article{LieStahnSullivan2022,
  author    = {Lie, Han Cheng and Stahn, Martin and Sullivan, Tim J.},
  title     = {Randomised one-step time integration methods for deterministic operator differential equations},
  series = {Calcolo},
  volume    = {59},
  journal   = {Calcolo},
  number    = {1},
  publisher = {Springer},
  address   = {Milano},
  issn      = {0008-0624},
  doi       = {10.1007/s10092-022-00457-6},
  pages     = {33},
  year      = {2022},
  abstract  = {Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.},
  language  = {en}
}
@article{LieStuartSullivan2019,
  author    = {Lie, Han Cheng and Stuart, A. M. and Sullivan, Tim J.},
  title     = {Strong convergence rates of probabilistic integrators for ordinary differential equations},
  series = {Statistics and Computing},
  volume    = {29},
  journal   = {Statistics and Computing},
  number    = {6},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0960-3174},
  doi       = {10.1007/s11222-019-09898-6},
  pages     = {1265 -- 1283},
  year      = {2019},
  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. ), 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.},
  language  = {en}
}