**Bayesian Inverse Problems**

## Department of Mathematics

# Bayesian inverse problems

Inverse problems are by definition problems that tolerate measurement and modelling errors poorly. Both types of errors are always present in real life applications. Several inverse problems were traditionally considered to be unsolvable, but during the last few decades, the understanding of inverse problems has been expanding significantly. The modelling errors have, however, been considered only lately.

The Bayesian, or statistical, framework for inverse problems is based on systematic modelling of all errors and uncertainties from the Bayesian viewpoint. This framework is significantly more tedious to build up than the traditional deterministic framework. As a reward, its potential to solve difficult inverse problems with high noise levels and serious model uncertainties is much higher.

The inverse problems research at the Department of Mathematics has strong ties with the Finnish Center of Excellence in Inverse Problems. The associated research consortium has been the leading group, among other topics, in the modelling of model uncertainties and especially the approximation error theory, which is an approach that provides computationally efficient implementations of inversion algorithms. In addition, the consortium is well known for the theory and applications of non-stationary inverse problems, that is, problems in which the unknown changes rapidly in time.

The applications include various biomedical, industrial and geophysical problems.

**Left:** Location of the submandibular nerve in the lower jawbone. This information is of importance in dental surgery since permanent paralysis will result if the nerve is damaged. The image was computed from data obtained with a standard dental X-ray system. The algorithms developed by the research consortium are currently implemented in production systems of the international market leader of dental X-ray systems.

**Right:** In Bayesian inverse problems, all unknowns are explicitly modelled as random variables, functions, processes and other different mathematical entities. These images form a statistical model describing possible (expected) conductivity distribution of a cross section of a head.

### Researchers at The University of Auckland

### Other collaborators

- Doctor Ville Kolehmainen

University of Eastern Finland, Department of Physics and Mathematics. - Arto Voutilainen

University of Eastern Finland, Department of Physics and Mathematics. - Aku Seppanen

University of Eastern Finland, Department of Physics and Mathematics. - Tanja Tarvainen

University of Eastern Finland, Department of Physics and Mathematics. - Janne Huttunen

University of Eastern Finland, Department of Physics and Mathematics. - Professor Erkki Somersalo

Case Western Reserve University, Department of Mathematics. - Daniela Calvetti

Case Western Reserve University, Department of Mathematics. - Professor Hikki Haario

Lappeenranta University of Technology, Department of Mathematics. - Professor Colin Fox

University of Otago, Department of Physics.

- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
- New Zealand Journal of Mathematics

**Programmes, Centres and Partners**

- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
- New Zealand Journal of Mathematics

**Programmes, Centres and Partners**