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

TL;DR

This paper establishes strong error bounds for randomized one-step time integration methods applied to deterministic operator differential equations, extending uncertainty quantification techniques to infinite-dimensional systems.

Contribution

It provides the first theoretical analysis of randomized time integration methods for infinite-dimensional deterministic differential equations, with weaker assumptions on local truncation errors.

Findings

Strong error bounds in Orlicz norms for random trajectories

Validation of randomized methods in infinite-dimensional settings

Weaker assumptions on local truncation error

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.