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.

中文翻译：

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混沌和几何数值积分中不确定性量化的随机时间步长概率方法

介绍了一种新的概率数值方法，用于量化由常微分方程 (ODE) 的时间积分引起的不确定性。与通过添加随机强迫项来随机化 ODE 求解器的经典策略不同，我们表明可以通过在经典时间积分器中引入合适的随机时间步长来获得 ODE 数值解的概率度量。这种内在随机化允许保持基本确定性积分器的几何属性，例如质量守恒、辛性或第一积分的守恒。导出弱收敛和均方收敛分析。我们还分析了所提出的随机时间步长方法的 Monte Carlo 估计器的收敛性，并表明通过重复采样获得的度量在均方意义上收敛，与样本数量无关。包括混沌哈密顿系统、化学反应和贝叶斯推理问题在内的数值例子说明了我们概率数值方法的准确性、鲁棒性和多功能性。