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
Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration
摘要：
提出了一种新的概率数值方法，用于量化由常微分方程（ODE）时间积分引起的不确定性。从经典的战略随机ODE求解器通过添加一个随机强迫项，我们证明了一个概率测度的数值解可以通过引入适当的随机时间步长在经典的时间积分器。这种内在的随机化允许基本确定性积分器的几何性质的守恒，例如质量守恒、辛性或首次积分守恒。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蒙特卡罗估计的收敛性，并证明了重复采样得到的测量在均方意义下的收敛性与样本的样本个数无关。包括混沌哈密顿系统，化学反应和贝叶斯推理问题的数值例子说明了我们的概率数值方法的准确性，鲁棒性和通用性。
英文摘要：
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.
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