Statistical aspects of non-linear inverse problems

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.