The course will consist of

• one week of on-campus activities at Linköping University (LiU), Campus Valla, from October 26 to 30, 2026;

• remote project work in teams (~80h) over around four weeks; and

• one final day for the student teams to present their projects.

5 ECTS (approx. 40h in person and 85h self-study/project time)

Send an email to jan.glaubitz@liu.se

• Inverse problems: The basics

• Regularization: Overcoming ill-posedness

• Bayes’ rule and other basics in probability theory

• Bayesian inverse problems

• Bayesian inference and uncertainty quantification

How can we reconstruct high-quality images from measurement data? How can we uncover biochemical reactions from experiments? And how can we quantify uncertainty in the predictions of mathematical models such as dynamical systems and neural networks?

These questions arise throughout e-Science and engineering whenever experimental or observational data are used to infer unknown parameters, states, or structures in mathematical models (e.g.,neural networks and dynamical systems). What they share is that they can all be formulated as inverse problems. The Bayesian approach provides a principled and versatile framework for addressing such inverse problems by modeling the unknowns probabilistically and combining prior information with data through Bayes’ rule. This perspective not only yields point estimates but also provides uncertainty quantification, a key ingredient for reliable inference and prediction in applications such as medical imaging, geophysics, climate modeling, weather forecasting, material science, and biochemical modeling.

This PhD course introduces students to the foundations of inverse problems, highlights their prevalence across e-Science domains, and presents modern computational strategies for solving them efficiently using state-of-the-art e-infrastructure. A key emphasis is on developing the statistical and computational skills needed to construct, analyze, and implement Bayesian inverse problem formulations, with particular attention to uncertainty quantification and its role in trustworthy scientific computing. The course thus addresses core methodological challenges central to e-Science.

After passing the course, students will be able to:

• formulate, analyze, and solve inverse problems

• be able to formulate and apply different regularization strategies—leveraging a priori (expert) knowledge to overcome ill-posedness

• apply Bayes’ rule to formulate inverse problems in a statistical setting

• argue about statistical modeling/data assumptions and formulate them

• use different inference techniques to solve Bayesian inverse problems

• be able to quantify uncertainty in estimated parameters

Basic analysis, basic linear algebra, and basic coding experience

Students are evaluated based on short reports and presentations on their project work.

TBA

Jan Glaubitz (jan.glaubitz@liu.se)
Assistant Professor
Department of Mathematics
Linköping University, Sweden