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

逆问题出现在科学、工程、技术和医学的各个领域。它们提供了一种从观测数据中提取知识和见解的系统和严格的方法。当这些数据与可以用数学模型描述的自然或工程系统的观测结果相对应时，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（PDEs）。有效的反问题求解方法利用这些性质是至关重要的。当反演参数是高（或无限）维时，当数学模型是由偏微分方程给出时，当人们对量化参数中的不确定性感兴趣时，这一点在许多应用中都很重要。该项目将系统地研究三个反问题的性质并开发算法，这三个反问题代表了由偏微分方程控制的广泛贝叶斯反问题：(1)具有空间（和时间）良好分离参数和观测位置的抛物型反问题；(2)具有丰富的测量数据集，但参数和观测点对应的位置没有良好分离的椭圆型Stokes流问题；(3)具有稀疏点测量值的双曲型问题。PI将从理论上研究这些问题，开发和分类结构利用方法来近似解决这些问题，并在开源软件库中实现这些方法。这三个原型问题都有重要的、与社会相关的现实世界的大规模类比。因此，从这三个模型问题中获得的任何算法或理论发现都将对这些大挑战逆问题有直接的好处。

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.