Inverse Problems Group

The inverse Problems group was appointed a Center of Excellence in Inverse Problems for the years 2006-2011 by the Academy of Finland. Other partners of the consortium are University of Helsinki, Helsinki University of Technology, Lappeenranta University of Technology and University of Oulu.

General information

Inverse problems are characterized by the property that the solutions are extremely sensitive to measurement and modelling errors. Ultimately, an infinity of unknown variables could have produced exactly the same set of measurements. The classical theory with which to tackle these problems is called regularization theory. While the traditional theory of inverse problems is centered in the "small noise, accurate models" case with known error norm, the vast majority are cases with "not-so-small-noise" and, furthermore, the error norms are not known and the models are invariably only approximations to physical reality.

In the statistical inversion paradigm both the measurement errors and the unknown variables are treated as random variables. The natural framework is then Bayesian statistics in which the strategy is to construct accurate models for the observations and the so-called prior model for the unknown variable. The central issue is not that the physical model is accurate in the deterministic sense, but that the discrepancies between the model and the reality are also modelled and these models are exploited.

In the above, our focus and contribution is targeted in computational issues as well as general topics in statistical inversion theory. Naturally, also the models for related physical systems are studied. Methodological applications and models include electrical impedance tomography, optical tomography, large-scale model-based control, limited angle tomography and ultrasound transmission problems. The end applications include biomedical, industrial and environmental applications such as ultrasound therapy (bloodless surgery) and dental radiology, process tomography and satellite imaging.

See also the book:
Kaipio JP, Somersalo E:
Statistical and Computational Inverse Problems,
Applied Mathematical Sciences 160, Springer-Verlag, ISBN: 0-387-22073-9, 2004.