We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon spatial and temporal refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensions. The method estimates the limiting exponential scaling using a single realization of the random variable, the large deviation minimizer. Finding this minimizer amounts to solving an optimization problem governed by a differential equation. The probability estimate becomes sharp when it additionally includes prefactor information, which necessitates computing the determinant of a second derivative operator to evaluate a Gaussian integral around the minimizer. We present an approach in infinite dimensions based on Fredholm determinants, and develop numerical algorithms to compute these determinants efficiently for the high-dimensional systems that arise upon discretization. We also give an interpretation of this approach using Gaussian process covariances and transition tubes. An example model problem, for which we also provide an open-source python implementation, is used throughout the paper to illustrate all methods discussed. To study the performance of the methods, we consider examples of stochastic differential and stochastic partial differential equations, including the randomly forced incompressible three-dimensional Navier-Stokes equations.
翻译:我们介绍并比较了针对带有小加性高斯噪声的随机微分方程中尖锐极端事件概率估计的计算技术。特别是,我们专注于可扩展的策略,即它们的效率不会因空间和时间精度而降低。为此,我们扩展了基于Laplace方法的算法,以估计无穷维中极端事件的概率。这种方法使用单次随机变量实现极限指数标度的估计,即大偏差极小化器。找到这个极小化器等价于解决一个由微分方程控制的优化问题。概率估计变得尖锐,当它另外包括系数信息时,这需要计算二阶导数算子的行列式,以评估围绕极小化器的高斯积分。我们在无限维中提出了一种基于Fredholm行列式的方法,并开发数值算法,以高效地计算这些行列式,以针对离散化后产生的高维系统。我们还使用Gaussian过程协方差和转换管的方法解释了这种方法。通过一个示例模型问题(我们也提供了开放源代码的Python实现),我们用它来说明本文中讨论的所有方法。为了研究方法的性能,我们考虑了随机微分方程和随机偏微分方程的示例,包括随机强制的不可压三维Navier-Stokes方程。