Statistical aspects of non-linear inverse problems

非线性反问题的统计方面

• 批准号: EP/Y030249/1
• 负责人: Richard Nickl
• 金额: $ 269.76万
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY030249%2F1
• 关键词: Statistical; aspects; non; linear; inverse

Statistical aspects of non-linear inverse problems The study of inverse problems forms an active field at the interface of applied and pure mathematics as well as the statistical, physical and biological sciences. Prototypical examples include parameter identification in partial differential equations (PDEs) but also tomography and data assimilation problems. While the theory can reach deep into delicate injectivity theorems and regularity theory for PDEs, applications feature prominently in various branches of applied sciences and more specifically in numerical analysis, imaging, statistics. These inference problems have recently drawn significant interest in the context of statistical data science, specifically through the development of Bayesian methods and related MCMC algorithms after seminal work by Andrew Stuart (2010). These can be used in high- or infinite-dimensional, non-linear, non-convex problems, and provide essential uncertainty quantification methods and 'error bars' for algorithmic outputs in complex inference tasks. Only very few rigorous statistical and computational guarantees for these algorithms are currently available, and whether such methods can be trusted in applications to the sciences and policy making remains unclear. The goal of this project is to close this gap and to build a satisfactory mathematical theory that explains both the empirical success and inherent limitations of Bayesian non-linear inversion methods in the context of 21st century data science.

非线性逆问题的统计方面的逆问题研究形成了应用和纯数学界面以及统计，物理和生物科学的活动。原型示例包括部分微分方程（PDE）中的参数识别，还包括层析成像和数据同化问题。尽管该理论可以深入研究PDE的精致的注射性定理和规律性理论，但应用在应用科学的各个分支中，更具体地说是在数值分析，成像，统计数据中的特征。这些推论问题最近引起了人们对统计数据科学背景的重大兴趣，特别是通过安德鲁·斯图尔特（Andrew Stuart）的开创性工作之后的贝叶斯方法和相关的MCMC算法的开发。这些可以用于高或无限的非线性，非线性，非凸问题，并为复杂的推理任务中的算法输出提供必要的不确定性量化方法和“误差栏”。当前可用的这些算法只有很少的严格统计保证和计算保证，并且在科学和政策制定应用程序中是否可以信任此类方法尚不清楚。该项目的目的是缩小这一差距，并建立令人满意的数学理论，该理论解释了在21世纪数据科学的背景下，贝叶斯非线性反演方法的经验成功和固有的局限性。