无限维 Bayes 反演理论与算法

Inverse problems constitute a significant area of mathematical research, with extensive applications across various engineering and technical fields such as medical imaging, seismic exploration imaging, image processing, and weather forecasting. Owing to the ill-posedness of inverse problems, the concept of regularization is introduced to solve these problems, resulting in an approximate estimation of the parameters. With the advancement of computational capabilities, people in fields like medical and exploration imaging are no longer satisfied with obtaining a reasonable estimate of the parameters to be estimated. Instead, they attempt to integrate empirical knowledge and uncertainty information of observational data to provide a complete characterization of the uncertainty of the parameters to be estimated. To achieve this goal, people transform inverse problems into Bayesian statistical inference problems, leading to the development of Bayesian inversion theory and numerical algorithms. Unlike classical statistical research, in inverse problem research, the parameters to be estimated and the observational data are connected by complex mathematical models (e.g., partial differential equations), thus necessitating the introduction of new ideas and mathematical theories. In this paper, we focus on the infinite-dimensional Bayesian inversion theory established for inverse problems and organize existing research work from aspects such as prior measure construction, Bayesian well-posedness, finite element discretization, statistical sampling algorithms, and statistical large-sample theory. The aim is to clarify the basic research ideas, core research issues, existing results and methods of infinite-dimensional Bayesian inversion methods, and potential future research directions.