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A novel probabilistic numerical method for quantifying the uncertaintyinduced by the time integration of ordinary differential equations (ODEs) isintroduced. Departing from the classical strategy to randomize ODE solvers byadding a random forcing term, we show that a probability measure over thenumerical solution of ODEs can be obtained by introducing suitable randomtime-steps in a classical time integrator. This intrinsic randomization allowsfor the conservation of geometric properties of the underlying deterministicintegrator such as mass conservation, symplecticity or conservation of firstintegrals. Weak and mean-square convergence analysis are derived. We alsoanalyse the convergence of the Monte Carlo estimator for the proposed randomtime step method and show that the measure obtained with repeated samplingconverges in the mean-square sense independently of the number of samples.Numerical examples including chaotic Hamiltonian systems, chemical reactionsand Bayesian inferential problems illustrate the accuracy, robustness andversatility of our probabilistic numerical method.

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