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Design and Sensitivity Analysis of Infinite-Dimensional Bayesian Inverse Problems

无限维贝叶斯反问题的设计与敏感性分析

基本信息

批准号：
2111044
负责人：
Alen Alexanderian
金额：
$ 26.5万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111044&HistoricalAwards=false
关键词：
Design Sensitivity Analysis Infinite Dimensional

项目摘要

Mathematical models of complex physical and biological systems play a crucial role in understanding real world phenomena and making predictions. Examples include models of weather systems, ocean circulation, contaminant transport, porous media flow, or spread of infectious diseases. Models governing complex systems typically include a large number of parameters that are needed for a full model specification. Typically, some of the model parameters are uncertain and need to be estimated using indirect measurements. This is done by solving an inverse problem that uses the model and measurement data to estimate the unknown parameters. Measurements are often scarce and noisy. Moreover, not all parameters can be estimated due to lack of data that informs them or the sheer computational cost associated with estimating all model parameters. This project makes fundamental contributions to parameter estimation and model-based prediction by establishing methods for assessing sensitivity of the solution of parameter estimation problems to additional model uncertainties and for making principled decisions on the choices of experiments one needs to conduct to obtain measurement data. The latter, i.e., the experimental design problem, is a crucial aspect of successful parameter estimation as it enables making judicious use of scarce experimental resources to obtain informative data. The PI will disseminate research results through peer-reviewed publications, organization of mini-symposia at international conferences, and release of open-source software. This project will support 1 graduate student each of the three years of the project. This research program focuses on Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameters. Examples include estimation of boundary conditions or coefficient functions in PDE models. Available measurement data are usually not sufficient to simultaneously inform all of the model parameters. Hence, the governing model typically contains parameters, herein referred to as auxiliary parameters, which are uncertain but must be specified for a complete model characterization necessary for an inverse problem formulation. An important question regarding such parameterized inverse problems is: what is the relative importance of the different auxiliary parameters to the solution of the inverse problem? This is addressed by developing a sensitivity analysis framework, called hyper-differential sensitivity analysis (HDSA), for large-scale Bayesian inverse problems. Another key aspect of successful parameter estimation is collection of informative experimental data. Physical or budgetary constraints often put severe limits on the amount of data that can be collected. Therefore, optimal data acquisition is crucial; this can be tackled through optimal experimental design (OED). This research program will bring about key advances in computational methods for PDE-based Bayesian inverse problems by developing methods for (i) analyzing the sensitivity of the solution of a Bayesian inverse problem to auxiliary parameters and (ii) fast computation of optimal experimental designs. The proposed methods achieve these goals through an intricate combination of rigorous methods from numerical analysis, randomized linear algebra, probability, and optimization.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.

复杂物理和生物系统的数学模型在理解现实世界现象和做出预测方面起着至关重要的作用。例如天气系统模型、海洋环流模型、污染物传输模型、多孔介质流动模型或传染病传播模型。管理复杂系统的模型通常包括完整模型规范所需的大量参数。通常，一些模型参数是不确定的，需要使用间接测量来估计。这是通过解决一个反问题来实现的，该反问题使用模型和测量数据来估计未知参数。测量结果往往很稀少，而且噪音很大。此外，并不是所有的参数都可以估计，因为缺乏数据来告知它们，或者与估计所有模型参数相关的纯粹的计算成本。该项目通过建立方法来评估参数估计问题的解决方案对额外模型不确定性的敏感性，并对获得测量数据所需进行的实验选择作出原则性决定，从而为参数估计和基于模型的预测作出了基本贡献。后者，即实验设计问题，是成功的参数估计的一个关键方面，因为它能够明智地利用稀缺的实验资源来获得信息数据。非政府组织将通过同行评议的出版物、在国际会议上组织小型研讨会和发布开放源码软件来传播研究成果。该项目将在项目的三年中每年资助一名研究生。本研究项目主要研究具有无限维参数的偏微分方程(PDE)所控制的贝叶斯反问题。例如，PDE模型中边界条件或系数函数的估计。现有的测量数据通常不足以同时告知所有模型参数。因此，控制模型通常包含参数，在此称为辅助参数，这些参数是不确定的，但是必须为反问题公式所需的完整模型表征而指定。关于这种参数化反问题的一个重要问题是：不同辅助参数对反问题求解的相对重要性是什么？这是通过为大规模贝叶斯反问题开发一个称为超微分灵敏度分析(HDSA)的灵敏度分析框架来解决的。成功的参数估计的另一个关键方面是收集信息丰富的实验数据。物质或预算方面的限制往往会对可收集的数据量造成严格限制。因此，最佳数据获取至关重要；这可以通过最佳试验设计(OED)来解决。该研究项目将通过开发(I)分析贝叶斯反问题解对辅助参数的敏感度和(Ii)快速计算最优试验设计的方法，在基于偏微分方程的贝叶斯反问题的计算方法方面取得重要进展。建议的方法通过数值分析、随机化线性代数、概率和优化等严格方法的复杂组合来实现这些目标。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。