TY  - JOUR
A1  - Lie, Han Cheng
A1  - Stuart, A. M.
A1  - Sullivan, Tim J.
T1  - Strong convergence rates of probabilistic integrators for ordinary differential equations
JF  - Statistics and Computing
N2  - 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.
KW  - Probabilistic numerical methods
KW  - Ordinary differential equations
KW  - Convergence rates
KW  - Uncertainty quantification
Y1  - 2019
U6  - https://doi.org/10.1007/s11222-019-09898-6
SN  - 0960-3174
SN  - 1573-1375
VL  - 29
IS  - 6
SP  - 1265
EP  - 1283
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - JOUR
A1  - Lie, Han Cheng
A1  - Stahn, Martin
A1  - Sullivan, Tim J.
T1  - Randomised one-step time integration methods for deterministic operator differential equations
JF  - Calcolo
N2  - 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.
KW  - Time integration
KW  - Operator differential equations
KW  - Randomisation
KW  - Uncertainty quantification
Y1  - 2022
U6  - https://doi.org/10.1007/s10092-022-00457-6
SN  - 0008-0624
SN  - 1126-5434
VL  - 59
IS  - 1
PB  - Springer
CY  - Milano
ER  -