Numerical optimization for large-scale experimental design of ill-posed inverse problems

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Inverse problems play a key role in a variety of fields such as computer vision, geophysics and medical imaging. The proposed work focuses on experimental design of ill-posed inverse problems; a field that has not received sufficient attention in the inverse problems community where the focus is usually on the analysis of the inverse problem given the data. Obviously, this already imposes restrictions on the quality of the possible solutions. On the other hand, the objective of the funded work is the study of an important pre-data acquisition question: How should the experiment be conducted to obtain optimal data given the physical constraints and available resources? Solutions to this question require techniques from numerical optimization, statistics and inverse problem theory. In particular, the problem of experimental design can be cast as a bilevel optimization problem that consists of two nested optimization problems. The proposed work is a study of design criteria and new numerical algorithms for the solution of the bilevel optimization problems that arise from them.This work will address the fundamental question of experimental design of ill-posed inverse problems. Such problems arise in the design of any practical experiment or instrumentation from geophysical and medical imaging to the production of better vision systems. This work will develop new criteria for the design and development of new algorithms that will enable its numerical implementation. The results of the research will be applied to electromagnetic imaging, a field that is routinely used in geophysics and medical physics. It will lead to better experiments that yield better images and as such, will assist in the decision making of geoscientists and and physicians.