Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

TL;DR

This paper introduces a probabilistic numerical method for uncertainty quantification in ODE integration by randomizing time steps, preserving geometric properties and providing convergence guarantees, with applications to chaotic systems and Bayesian inference.

Contribution

It proposes a novel random time step approach that maintains geometric properties and offers convergence analysis, advancing probabilistic ODE solvers.

Findings

The method preserves mass, symplecticity, and first integrals.

Convergence of the measure is independent of sample size.

Numerical examples demonstrate robustness and versatility.

Abstract

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time-steps in a classical time integrator. This intrinsic randomization allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean-square convergence analysis are derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean-square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.