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 temporal and possibly spatial refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensional path space. 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 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：我们介绍并比较了在带有小加性高斯噪声的随机微分方程中，计算尖锐极端事件概率估计的计算技术。特别地，我们关注可伸缩的策略，即其效率不会因时间及可能的空间细化而降低。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路径空间。该方法利用随机变量的一次实现——即大偏差极小化子——来估计极限指数标度。寻找该极小化子等价于求解一个由微分方程支配的优化问题。当进一步包含前因子信息时，概率估计变得尖锐，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，以高效计算离散化产生的高维系统中的这些行列式。我们还利用高斯过程协方差和转移管道对此方法进行了解释。论文中通过一个示例模型问题（我们提供其开源Python实现）来说明所讨论的所有方法。为研究各方法的性能，我们考虑了随机微分方程和随机偏微分方程的例子，包括随机强迫不可压三维纳维-斯托克斯方程。