Statistical aspects of non-linear inverse problems

非线性反问题的统计方面反问题的研究在应用数学和纯数学以及统计、物理和生物科学的界面上形成了一个活跃的领域。典型的例子包括偏微分方程（PDEs）中的参数辨识，以及断层扫描和数据同化问题。虽然该理论可以深入到精细的注入性定理和偏微分方程的正则性理论，但其应用在应用科学的各个分支中都很突出，更具体地说，在数值分析、成像、统计中。这些推理问题最近在统计数据科学的背景下引起了极大的兴趣，特别是在Andrew Stuart（2010）的开创性工作之后，通过贝叶斯方法和相关MCMC算法的发展。这些可以用于高维或无限维，非线性，非凸问题，并为复杂推理任务中的算法输出提供必要的不确定性量化方法和“误差条”。目前对这些算法只有很少的严格的统计和计算保证，而且这些方法在科学和政策制定中的应用是否值得信任仍然不清楚。该项目的目标是缩小这一差距，并建立一个令人满意的数学理论，以解释贝叶斯非线性反演方法在21世纪数据科学背景下的经验成功和固有局限性。

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.