# Strong convergence rates of probabilistic integrators for ordinary differential equations

## TL;DR

This paper analyzes the convergence rates of probabilistic integrators for ordinary differential equations, showing they match deterministic methods under certain conditions, thus enabling reliable uncertainty quantification.

## Contribution

It extends convergence analysis to probabilistic integrators with relaxed assumptions, demonstrating their mean-square convergence rates match deterministic methods.

## Key findings

- Probabilistic integrators achieve the same convergence rates as deterministic ones.
- Convergence holds for state-dependent, non-Gaussian, and non-centred random perturbations.
- Results apply to high-order integrators for Lipschitz flows and Euler methods for dissipative fields.

## Abstract

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model 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.\ (\textit{Stat.\ Comput.}, 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.