Classification of Methods for Bayesian Inverse Problems Governed by Partial Differential Equations

逆问题出现在科学、工程、技术和医学的所有领域。它们提供了一种从观测数据中提取知识和洞察力的系统和严格的方法。当这些数据对应于可以用数学模型描述的自然或工程系统的观测值时，反问题的性质和结构取决于这些模型的性质，这些模型通常涉及偏微分方程(PDE)。有效的反问题求解方法利用这些性质是至关重要的。特别是当反演参数是高维(或无限)维时，当数学模型由偏微分方程给出时，以及当人们对量化参数中的不确定性感兴趣时，这在许多应用中是重要的。本项目将系统地研究三个逆问题的性质并开发算法，这三个逆问题代表由偏微分方程控制的一大类贝叶斯反问题：(1)具有空间(和时间)良好分离的参数和观测位置的抛物型反问题，(2)椭圆型斯托克斯流问题，其具有丰富的测量数据集，并且对应于参数和观测的位置没有很好地分离，(3)具有稀疏点测量的双曲问题。PI将从理论上研究这些问题，开发和分类结构开发方法来近似解决这些问题，并在开放源码软件库中实现这些方法。所有这三个原型问题都有重要的、与社会相关的现实世界、大规模类似物。因此，针对这三个模型问题获得的任何算法或理论发现都将对这些重大挑战的反问题产生直接的好处。

Inverse problems emerge in all areas of science, engineering, technology, and medicine. They provide a systematic and rigorous way to extract knowledge and insight from observational data. When this data corresponds to observations of natural or engineered systems that can be described by mathematical models, the properties and structure of the inverse problem depend on the properties of these models, which commonly involve partial differential equations (PDEs). It is crucial that efficient inverse problem solution methods exploit these properties. This is in particular the case when the inversion parameters are high (or infinite) dimensional, when the mathematical models are given by PDEs, and when one is interested in quantifying the uncertainty in the parameters, as is important in many applications.This project will systematically study properties and develop algorithms for three inverse problems that are representative of a wide class of Bayesian inverse problems governed by PDEs: (1) a parabolic inverse problem with spatially (and temporally) well-separated parameter and observation locations, (2) an elliptic Stokes flow problem for which a rich set of measurement data are available and the locations corresponding to parameters and observations are not well-separated, and (3) a hyperbolic problem with sparse point measurements. The PI will study these problems theoretically, develop and classify structure-exploiting methods to approximate their solutions, and implement these methods in an open-source software library. All three prototype problems have important and societally relevant real-world, large-scale analogues. Thus, any algorithmic or theoretical findings obtained for the three model problems will have immediate benefit for these grand challenge inverse problems.