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：我们引入并比较了在带有小加性高斯噪声的随机微分方程中，用于尖峰极端事件概率估计的计算技术。特别地，我们关注可扩展的策略，即其效率不会因空间和时间细化而降低。为此，我们将基于拉普拉斯方法估计极端事件概率的算法扩展到无限维。该方法利用随机变量的单次实现——大偏差极小化子——来估计极限指数标度。寻找该极小化子需要求解一个由微分方程支配的优化问题。当额外包含前因子信息时，概率估计变得精确，这需要计算二阶导数算子的行列式以评估极小化子周围的高斯积分。我们提出了一种基于弗雷德霍姆行列式的无限维方法，并开发了数值算法来有效计算离散化产生的高维系统的这些行列式。我们还使用高斯过程协方差和过渡管给出了该方法的解释。本文使用一个示例模型问题（我们也为其提供了开源的Python实现）来阐明所有讨论的方法。为研究所提出方法的性能，我们考虑了随机微分方程和随机偏微分方程的例子，包括随机受迫的三维不可压缩纳维-斯托克斯方程。