## 1 PhD Position within the FWF Project 'Data-driven and problem-oriented choice of the regularization space'

Organization:
Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences (ÖAW)
Email:
sergei.pereverzyev$oeaw.ac.at Job Description: PhD position (f/m) in the Inverse Problems Group at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences we are advertising 1 Doctoral position for a term of up to 3 years at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences within the FWF-project Data-driven and problem-oriented choice of the regularization space. We search for interested candidates with a strong background in one or more of the following fields: functional analysis, inverse problems, optimization and optimal control, numerical analysis, regularization theory, learning theory. Applications with a motivation letter, curriculum, a description of research interests, up to 3 letters of recommendation, and final master grades should be submitted electronically. To the successful candidates will be offered research positions up to 3 years with the salary defined according to the FWF guidelines. The starting date is currently planned on June 1, 2013 (negotiable). Application deadline: April 30, 2013, or until filled. Enquiries regarding the positions and the applications should be directed to Prof. Dr. Sergei Pereverzyev (sergei.pereverzyev$oeaw.ac.at).

Summary of the project FWF Project Data-driven and problem-oriented choice of the regularization space
Regularization is an approach to approximate reconstruction of the unknown functional dependency from available noisy data. This approach is often based on a compromise between the attempt to fit given data and the desire to reduce complexity of a data fitter.
Starting from the pioneering work of Tikhonov and Wahba, Kimeldorf in the the mid-sixties a huge body of the regularization theory has been built around the issue of choosing the regularization parameter, which is just a one-number choice. At the same time, the most recent trend in the regularization theory has led to a new line of research of the adaptive choice of the regularization space, with still many open questions.
The current project aims at comprehensive theoretical analysis of this issue and extensive numerical studies. In particular, two research directions will be explored. One of them leads to the multi-penalty regularization (MPR), where only the first theoretically justified results on adaptive selection of multiple regularization parameters have been obtained. Since the idea of a MPR usage is attractive as it opens a possibility of combining several approximation tasks such as predictions by interpolation and by extrapolation, one argues that under some assumptions, regularization in an adaptively chosen space can be reduced to a multi-penalty regularization with a component-wise penalization.
Another research direction appears on the border between regularization theory and meta-learning. In rough terms, a meta-learning based regularization means that the instances of a regularization method are chosen from experience with this method in similar applications. Such frame covers several recently proposed approaches, where only a single instance is extracted from the experience. But it seems to be more promising to use the experience for finding a rule for choosing regularization instances in dependence on features (meta-features) of a particular application. It appears that such approach has not been systematically studied so far in the regularization theory and the current project aims to shed the light on this promising but as of yet not researched area.
Both above mentioned project directions may, in the future, form one of mainstreams in the regularization theory and can play a fundamental role for some practical applications, such as geomathematics and diabetes technology.

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