Uncertainty Quantification and Inverse Problems

Inverse problems and uncertainty quantification are closely related fields essential in many scientific and engineering disciplines. Inverse problems involve deducing unknown causes or system parameters from observed effects (data), often encountered in areas like medical imaging or geophysics. These problems are challenging due to issues like non-uniqueness and instability. Uncertainty quantification complements this by assessing and managing the uncertainty in models and simulations, crucial for making reliable predictions. Together, these disciplines focus on developing robust methods for understanding and predicting complex systems despite inherent uncertainties and limited data.

In this course, we will cover the following topics:

- basics from probability theory and functional analysis

- inverse problems and optimization

- Bayesian (regularization of) inverse problems

- measures of information and uncertainty; Bayesian experimental design

- Gaussian process regression

- numerical integration in high dimensions and sampling (Monte Carlo, importance sampling, MCMC)

Literature

• H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems, Kluwer, 1996 /2000.
• A. Kirsch: An introduction to the mathematical theory of inverse problems, Springer 2011 (2. Auflage).
• J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer 2005.
• M. Dashti and A.M.Stuart. The Bayesian Approach to Inverse Problems. Handbook of Uncertainty Quantification, Springer, 2015.