Efficient Algorithms Related to and Beyond the Large Deviation Technique

基于偏微分方程的数学建模方法在工程和科学应用中得到了广泛的应用，它已成为人类理解人类活动和技术发展所产生的大量现象的最重要工具之一。随机偏微分方程通过考虑不确定性对现实中普遍存在的偏微分方程进行了推广。在这个项目中，我们的重点是随机偏微分方程中罕见事件的模拟和量化，这些事件可以模拟一些重要的现象，如气候的政权变化，流氓海浪，异常天气等，这些事件虽然很少发生，但对我们的生活有重大影响。该项目的主要目标是开发有效的数值算法来捕获无限维系统中的罕见事件。我们将把偏微分方程数值解的技术，如有限元法、降基法等（对于时空维度）与大偏差理论、统计学和深度学习（对于随机维度）的思想相结合。当大偏差原理适用时，我们将考虑非局部变分问题的数值解，以寻求最可能的事件。该算法将在有限元法和变分法的框架下进行开发和分析。当大偏差原理不适用时，我们将开发一种策略，将深度学习的降阶建模与生成模型无缝耦合，并在此基础上构建更通用的交叉熵方法用于罕见事件模拟。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical modeling methods based on partial differential equations are widely used in engineering and scientific applications, which have been one of the most important tool for mankind to understand a large variety of phenomena originating from human activity and technological development. The stochastic partial differential equations generalize the partial differential equations by taking into account the uncertainty, which is ubiquitous in reality. In this project, we focus on the simulation and quantification of rare events in stochastic partial differential equations that can model some important phenomena such as regime change in climate, rogue ocean waves, abnormal weather, etc, which may occur rarely but have major impact on our life.The main goal of this project is to develop efficient numerical algorithms to capture rare events in infinite dimensional systems. We will integrate the techniques for numerical solution of partial differential equations, such as finite element method, reduced basis method, etc, (for the space-time dimension), with the ideas from large deviation theory, statistics, and deep learning (for the random dimension). When the large deviation principle is applicable, we will consider numerical solution of a nonlocal variational problem to seek the most probable event. The algorithm will be developed and analyzed in the framework of finite element method and calculus of variation. When the large deviation principle is not applicable, we will develop a strategy to seamlessly couple the reduced-order modeling and the generative models from deep learning, based on which a more general cross entropy method will be constructed for rare event simulations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.