LEAPS-MPS: Dynamical Parameter Estimation for Hydrodynamic Equations

LEAPS-MPS：流体动力学方程的动力学参数估计

• 批准号: 2213363
• 负责人: Vincent Martinez
• 金额: $ 23.89万
• 依托单位: CUNY Hunter College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2213363&HistoricalAwards=false
• 关键词: LEAPS; MPS; Dynamical; Parameter; Estimation

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).A fundamental feature in the modeling of physical phenomena is the presence of parameters that encode various relevant properties. In the study of fluid flows, parameters arise that capture the ability of the fluid to generate friction by shearing against itself or quantify how temperature differences within the fluid can drive its motion, such as when the sun heats the ocean and a flame heats a pot of water, or more generally represent external sources of energy, such as disturbances in the atmosphere due to large-scale disruptive volcanic eruptions and anthropogenic greenhouse gas effects. The role of these parameters is crucial in the understanding of turbulence, climate dynamics, and other engineering applications. Their precise determination is obtained empirically through the collection of data. Often the data collected is incomplete or corrupted by noise that may arise from the measurement procedure, or the devices used. This project systematically addresses the issue of the accurate estimation of parameters in the context of fluid flows, through the mathematical analysis and computational study of a data-driven approach geared to identify conditions under which recovery of the parameters can be provably guaranteed in several suitably idealized practical situations. The project also builds infrastructure for harnessing the data revolution through course redevelopment, provides training and mentoring opportunities for undergraduate and graduate students, and fosters a robust and diverse mathematical culture through a peer mentorship program. The project will carry out a mathematical study of a data-driven dynamical algorithm for recovering unknown parameters arising in partial differential equations of hydrodynamics via a paradigm of feedback control. In this approach, observations on the system are directly inserted into the prognostic equation as an exogenous term that enforces convergence towards the observations. Several practical scenarios of parameter estimation are conceived and systematically analyzed, which include the simultaneous recovery of multiple parameters and the use of various modes of observations. A key property underpinning the success of this strategy is the existence of a known nonlinear mechanism found in a large class of dissipative systems that allows small scale information to be asymptotically enslaved to large scale information. The application of this property to parameter estimation is novel. As a result, this study will shed light on the role of this mechanism for parameter estimation in contexts beyond fluid flow and provide a comprehensive exploration of the robustness and limitations of this approach.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.

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。物理现象建模的一个基本特征是存在对各种相关性质进行编码的参数。在流体流动研究中，出现了一些参数，这些参数捕捉到了流体通过对自身剪切而产生摩擦的能力，或者量化了流体内部的温差如何驱动其运动，例如当太阳加热海洋和火焰加热一壶水时，或者更一般地表示外部能源，例如大规模破坏性火山喷发和人为温室气体效应造成的大气扰动。这些参数的作用对于理解湍流、气候动力学和其他工程应用至关重要。它们的精确测定是通过收集数据来获得的。通常，收集的数据不完整或被测量程序或使用的设备可能产生的噪声破坏。该项目通过对数据驱动的方法进行数学分析和计算研究，系统地解决了在流体流动的情况下准确估计参数的问题，该方法旨在确定在几种适当的理想化的实际情况下可以证明能够保证恢复参数的条件。该项目还通过课程再开发为利用数据革命建立基础设施，为本科生和研究生提供培训和指导机会，并通过同行指导计划培养强大和多样化的数学文化。该项目将对一种数据驱动的动态算法进行数学研究，该算法用于通过反馈控制范例恢复流体动力学偏微分方程组中出现的未知参数。在这种方法中，对系统的观测被直接插入到预测方程中，作为一个外生项，强制向观测收敛。对参数估计的几种实际情况进行了设想和系统的分析，其中包括同时恢复多个参数和使用各种观测模式。这一策略成功的一个关键特性是在一大类耗散系统中存在一种已知的非线性机制，该机制允许小规模信息被渐近地奴役于大规模信息。这一性质在参数估计中的应用是新颖的。因此，这项研究将阐明这一参数估计机制在流体流动以外的环境中的作用，并全面探索这种方法的稳健性和局限性。该奖项反映了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。