Efficient Algorithms Related to and Beyond the Large Deviation Technique

Louisiana State University, Baton Rouge LA

Investigators

Abstract

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.