Introduction to Bayesian inverse problems

• Formulate and model inverse problems.
• Apply Monte Carlo methods and evaluate their convergence.
• Discretize continuous random variables using Karhunen-Loéve expansion.
• Apply statistical approaches to solve inverse problems (e.g., maximum likelihood estimation).
• Describe the modeling and computational elements of the Bayesian approach to inverse problems.
• Formulate different types of noise models and priors (e.g., conjugate priors).
• Describe the relevant strategies and implement numerical methods for Bayesian computations to practical problems (e.g., Markov chain Monte Carlo).
• Interpret and understand uncertainty quantification (UQ) results.
• Use the software package CUQIpy.

This short course will present fundamentals of Bayesian inversion. The idea is to cover both computational methods (e.g., Monte Carlo,discretization of random fields, Markov chain Monte Carlo) and theoretical aspects (e.g., basic proofs, convergence properties, well-posedness).

We start by presenting an overall introduction to the Bayesian framework and reviewing probability theory. This is performed with a very simple example that illustrates what is the Bayesian approach and why is it good for us. Next, we explore Monte Carlo methods for the simulation of random variables and estimation of expectations. After these introductory topics, we discuss the main component of the course, where we explore Bayesian approaches for inverse problems. The idea is to formulate elemental inverse problems and present statistical approaches to solve them. We will talk about likelihoods/noise models, and prior distributions as mechanism of regularization. The solution of the Bayesian inverse problem, given in terms of posterior statistics, is computed via Markov chain Monte Carlo sampling.

We describe how to formulated and solve inverse problems in a statistical setting by means of the software package CUQIpy, and we describe how the user can adjust the computational methods in this package.

The course aims at giving a hands-on experience, i.e., the student will learn how to apply a method and how to interpret the associated results. Therefore, lectures explaining the theory will be followed by exercise sessions.

Experience with probability theory (e.g., 02405), inverse problems (e.g., 02624), numerical computations (e.g., 02601), functional analysis (e.g., 01715) would be an advantage.

Lectures and exercises with theory and computations. Project work towards the end of the course.