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

正如在过去的飓风如卡特里娜(2005)、艾克(2008)、桑迪(2012)以及2017年一系列飓风中观察到的那样，风暴潮和降雨造成的洪水对沿海社区造成了巨大破坏。为应对未来的飓风冲击，需要能够预测由于洪水激增而导致的内陆洪水的特征，最重要的是水位、洋流和洪水淹没的程度。准确预测涌浪需要确定与沿海地区物理特征有关的模型的关键输入，这些模型获取和及时演变的成本很高。该研究项目致力于使用新的数学技术来量化和减少这些关键输入中的不确定性，以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，该项目专注于构建、分析和实施随机反问题的数值方法，该反问题通过融合观测数据和高保真数学模型来执行复杂物理系统的科学和工程推理和预测。这项研究广泛适用于依赖于高维参数空间的复杂的、随时间演化的物理系统。本研究使用的数学领域包括计算测度论、微分几何、泛函分析、概率论和数值分析。向更广泛的科学界传播结果将部分通过开发和实施公共领域代码中的计算算法来实现，这些算法应用于最先进的风暴潮模型。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.