Design and Sensitivity Analysis of Infinite-Dimensional Bayesian Inverse Problems

复杂的物理和生物系统的数学模型在理解现实世界的现象和做出预测方面起着至关重要的作用。例子包括天气系统、海洋环流、污染物运输、多孔介质流动或传染病传播的模型。控制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。这是通过求解一个逆问题来完成的，该问题使用模型和测量数据来估计未知参数。测量结果往往是稀缺和嘈杂的。此外，并非所有参数都可以估计，因为缺乏数据，或者与估计所有模型参数相关的纯粹计算成本。该项目通过建立评估参数估计问题解对附加模型不确定性的敏感性的方法，以及对获得测量数据所需进行的实验选择做出原则性决策，对参数估计和基于模型的预测做出了根本性贡献。后者，即实验设计问题，是成功参数估计的一个关键方面，因为它可以明智地利用稀缺的实验资源来获得信息数据。PI将通过同行评审的出版物、在国际会议上组织小型专题讨论会以及发布开源软件来传播研究成果。该项目将在项目的三年里每年资助一名研究生。本研究项目主要研究由具有无限维参数的偏微分方程（PDEs）控制的贝叶斯反问题。例子包括PDE模型中边界条件或系数函数的估计。可用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，这里称为辅助参数，这些参数是不确定的，但必须指定，以完成反问题公式所必需的完整模型表征。关于这种参数化逆问题的一个重要问题是：不同的辅助参数对逆问题的解的相对重要性是什么？这是通过开发一个灵敏度分析框架来解决的，称为超微分灵敏度分析（HDSA），用于大规模贝叶斯反问题。成功的参数估计的另一个关键方面是收集翔实的实验数据。物理上或预算上的限制往往严重限制了可以收集的数据量。因此，最佳数据采集是至关重要的；这可以通过优化实验设计（OED）来解决。本研究项目将在基于偏微分方程的贝叶斯反问题的计算方法方面取得关键进展，包括(1)分析贝叶斯反问题解对辅助参数的敏感性和(2)优化实验设计的快速计算。所提出的方法通过数值分析、随机线性代数、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。

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.