Bayesian Inverse Problems

Inverse Problems are a fascinating branch of applied mathematics with strong connections to both pure mathematical areas such as (functional) analysis and probability theory as well as the applied disciplines numerical analysis, optimization, statistics and machine learning. At the same time, inverse problems arise in essentially all the engineering and physical sciences. In a nutshell, inverse problems consist of inferring a quantity of interest or unknown from one or more noisy indirect observations. As such problems are typically ill-posed in the sense that they violate at least one of Hadamard's three requirements, specialized mathematical methods are required for their solution.

Until recently, the field was dominated by deterministic formulations and variational regularisation methods, which yield a solution approximation subject to additional constraints, typically involving smoothness. The Bayesian formulation, by contrast, is probabilistic, and yields a probability distribution for the unknown rather than a single estimate, allowing one to incorporate prior information as well as statistics on observational noise in a natural way. The course will cover the mathematical background of the Bayesian formulation of inverse problems and introduce the most important solution methods.  

The course is designed to be understandable for master's students. Previous knowledge of an inverse problems lecture is helpful, but not necessary.

The course language (English or German) will be determined in the first lecture depending on the audience.