Randomised one-step time integration methods for deterministic operator differential equations

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.\ (\textit{Stat.\ Comput.}, 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.

28 pages

References in corpus (4)

• Data Assimilation: A Mathematical Introduction
• A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems
• Bayesian Numerical Methods for Nonlinear Partial Differential Equations
• Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

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