Computational Bayesian Inverse Problems

Inverse problems constitute an important and active field of research of ap- plied mathematics, with application areas in medical imaging, geophysics, engineering, remote sensing and environmental sciences, life sciences and many more. In a nutshell, mathematical modeling aims at building models that explain and predict consequences of given causes; in inverse problems, the objective is to identify and quantify the causes of an observed conse- quence. Characteristic to inverse problems is that the observations can be explained by many different causes (non-uniqueness), and that small errors in observations may propagate to huge errors in the solution of the problem (instability). To address these problems, sophisticated techniques have been developed, including classical regularization methods and Bayesian statisti- cal techniques.

The focus of this course will be on Bayesian statistical methods in inverse problems. In this framework, the inverse problems are recast in a form of statistical inference problems, in which the unknowns are modelled as random variables, and the indirect observations are used to infer on the distribution of these variables. The distribution expresses the level of information about the unknowns, quantifying also the uncertainty. The latter point connects Bayesian inverse problems with the field of uncertainty quantification (UQ), an area that is increasingly important in the era of big data.

As prerequisites, the students should be familiar with basic linear algebra and numerical methods; familiarity with probability and statistics can be helpful. Examples and exercises involve the use of Matlab.

BIBLIOGRAPHY
- Jari Kaipio and Erkki Somersalo: Statistical and Computational Inverse Problems. Springer Verlag, 2004.
- Daniela Calvetti and Erkki Somersalo: Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing. Springer Verlag, 2007.

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• BCAM-UPVEHU Courses 2014-15 SOMERSALO