Collaborative Research: Construction and Analysis of Numerical Methods for Stochastic Inverse Problems with Application to Coastal Hydrodynamics

从过去的卡特里娜（2005年）、艾克（2008年）、桑迪（2012年）以及2017年飓风季的一系列飓风中可以观察到，风暴潮和降雨导致的洪水给沿海社区造成了巨大的破坏。为未来飓风的影响做准备需要预测内陆洪水特征的能力，最重要的是水位、水流和淹没程度。对风暴潮的准确预测需要确定与沿海地区物理特征相关的模型的关键输入，这些模型的获取成本很高，而且会随着时间的推移而变化。本研究项目的重点是量化和减少这些关键输入的不确定性，使用新颖的数学技术从现有数据中提取信息，并帮助设计未来的数据收集工作。更具体地说，该项目侧重于构建、分析和实现由观测数据和高保真数学模型融合定义的随机逆问题的数值方法，以执行复杂物理系统的科学和工程推断和预测。本研究广泛应用于依赖于高维参数空间的复杂、时变的物理系统。本研究运用的数学领域包括计算测量理论、微分几何、泛函分析、概率论和数值分析。将结果传播给更广泛的科学界，将部分地通过开发和实施公共领域代码中的计算算法来完成，这些算法将应用于最先进的风暴潮模型。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。

As observed in past hurricanes such as Katrina (2005), Ike (2008), Sandy (2012), and the sequence of hurricanes over the 2017 season, flooding due to storm surge and rainfall causes tremendous damage to coastal communities. Preparing for future hurricane impacts requires the ability to predict characteristics of inland flooding due to surge, most importantly water levels, currents, and extent of inundation. Accurate predictions of surge require determining critical inputs to models related to the physical characteristics of coastal regions that are expensive to obtain and evolve in time. This research project focuses on quantifying and reducing uncertainties in these critical inputs using novel mathematical techniques to both extract information from existing data and aid in the design of future data collection efforts.More specifically, this project focuses on the construction, analysis, and implementation of numerical methods for a stochastic inverse problem defined by the melding of observational data and high-fidelity mathematical models to perform scientific and engineering inference and prediction for complex physical systems. This research applies broadly to complex, time-evolving physical systems depending on high dimensional parameter spaces. Mathematical areas utilized in this research include computational measure theory, differential geometry, functional analysis, probability theory, and numerical analysis. Dissemination of results to the broader scientific community will be accomplished in part by the development and implementation of computational algorithms in public domain code that are applied to a state-of-the-art storm surge model.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.