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

基本信息

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

项目摘要

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.

科学，工程，技术和医学的所有领域都出现了反问题。它们提供了一种系统，严格的方式来从观察数据中提取知识和见解。当此数据对应于可以通过数学模型描述的天然或工程系统的观察，逆问题的属性和结构取决于这些模型的属性，这些模型通常涉及部分微分方程（PDES）。至关重要的是，有效的逆问题解决方案方法利用这些特性。 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在空间（和时间上）分离良好的参数和观察位置的逆问题，（2）一个椭圆形的stokes流量问题，可以为其提供一组丰富的测量数据，并且与参数和观察相对应的位置并未得到很好的分离，以及（3）具有稀疏点测量的双重核问题问题。 PI理论上将研究这些问题，开发和分类结构开发方法以近似其解决方案，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且与社会相关的现实世界，大规模类似物。因此，对于三个模型问题，获得的任何算法或理论发现将对这些巨大的挑战反向问题具有直接的好处。

项目成果

期刊论文数量（8）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

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
影响因子：
0
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
通讯作者：
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.
影响因子：
0
作者：
Chen Li-;G. Stadler
通讯作者：
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
影响因子：
0
作者：
Shanyin Tong;E. Vanden-Eijnden;G. Stadler
通讯作者：
Shanyin Tong;E. Vanden-Eijnden;G. Stadler

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Georg Stadler其他文献

Constraining Earth's nonlinear mantle viscosity using plate-boundary resolving global inversions.

使用板块边界解析全局反演来约束地球的非线性地幔粘度。

DOI：
10.1073/pnas.2318706121
发表时间：
2024
期刊：
Proceedings of the National Academy of Sciences of the United States of America
影响因子：
11.1
作者：
Jiashun Hu;J. Rudi;M. Gurnis;Georg Stadler
通讯作者：
Georg Stadler