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

偏微分方程治理贝叶斯反问题方法的分类

• 批准号: 1723211
• 负责人: Georg Stadler
• 金额: $ 18万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1723211&HistoricalAwards=false
• 关键词: Classification; Methods; Bayesian; Inverse; Problems

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.

逆问题出现在科学、工程、技术和医学的所有领域。它们提供了一种系统和严格的方法来从观测数据中提取知识和见解。当这些数据对应于可以通过数学模型描述的自然或工程系统的观测时，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属性是至关重要的。当反演参数高时，情况尤其如此（或无限）维，当数学模型由偏微分方程给出，当一个人有兴趣量化参数的不确定性，这在许多应用中是很重要的。这个项目将系统地研究属性和开发算法的三个反问题，这是广泛的一类贝叶斯反问题的偏微分方程的代表：（1）具有空间（和时间）分离的参数和观测位置的抛物型逆问题，（2）椭圆斯托克斯流问题，对于该椭圆斯托克斯流问题，可获得丰富的测量数据集，并且对应于参数和观测的位置没有分离，以及（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方案，并在开源软件库中实现这些方法。 所有这三个原型问题都有重要的和社会相关的现实世界，大规模的类似物。因此，对于这三个模型问题获得的任何算法或理论发现将对这些大挑战逆问题具有直接的益处。

项目成果

Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs

由偏微分方程控制的线性反问题的不可约不确定性下的最优实验设计

• DOI: 10.1088/1361-6420/ab89c5
• 发表时间: 2020
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Koval, Karina;Alexanderian, Alen;Stadler, Georg
• 通讯作者: Stadler, Georg

Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies

具有粘塑性流变学的斯托克斯流地球动力学模型的高级牛顿方法

• DOI: 10.1029/2020gc009059
• 发表时间: 2020
• 期刊: Geosystems
• 作者: Rudi, Johann;Shih, Yu‐hsuan;Stadler, Georg
• 通讯作者: Stadler, Georg

A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs

• DOI: 10.1088/1361-6420/aaf129
• 发表时间: 2018-08
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: B. Crestel;G. Stadler;O. Ghattas

Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case

• DOI: 10.1137/18m1181419
• 发表时间: 2018-04
• 期刊: SIAM J. Control. Optim.
• 作者: Chen Li-;G. Stadler

Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis

• DOI: 10.2140/camcos.2021.16.181
• 发表时间: 2020-07
• 期刊: ArXiv
• 作者: Shanyin Tong;E. Vanden-Eijnden;G. Stadler