The course offers an introduction to the subject of statistical
inversion, where, in its most basic form, the goal is to study how to
estimate model parameters from data. We will introduce mathematical
concepts and computational tools for systematically treating these
inverse problems in a Bayesian framework, including an assessment of
how uncertainties affect the solution. In the first part of the
course, we will study the Bayesian framework for finite-dimensional
inverse problems. While the first part will introduce some
machine-learning ideas, the second part will address how machine
learning is impacting, and has the potential to impact further on, the
subject of inverse problems. In the final part of the course, we will
generalize the Bayesian inverse problem theory to a Banach space
setting and discuss sampling strategies for accessing the Bayesian
posterior.

Topics covered include:
- Bayesian Inverse Problems and Well-Posedness
- The Linear-Gaussian Setting
- Optimization Perspective on Bayesian Inverse Problems
- Gaussian Approximation
- Markov Chain Monte Carlo 
- Blending Inverse Problems and Machine-Learning 
- Bayesian Inversion in Banach spaces (if time permits)