Inverse problems generally speaking determine causes for desired orobserved effects. An example for this is the reconstruction ofstructures inside the human body from measurements outside, as it isdone in medical imaging. In particular, in electrical impedancetomography one measures (via electrodes) the voltage pattern on thebody surface corresponding to different imposed current patterns onthe body surface. These patterns are crucially influenced by theconductivity distribution inside the body, so here the conductivitydistribution is the cause for the observed voltage-current effects onthe surface. Inverting this cause-to-effect map one can recover theconductivity distribution inside, which by assigning typicalconductivity values for e.g., lungs, heart, benign and malignanttissue, etc., gives an image of the interior of the body.
Inverseproblems have many other applications ranging from thecharacterization of materials via the detection of defects insidedevices to calibration of models in biology as well as economic andand social sciences. Computational methods for solving inverseproblems usually rely on some kind of inversion of the mentionedcause-to-effect map, which is also called forward operator.
However,this forward operator is often compuationally quite expensive toevaluate or might even not be well-defined. In such cases it can helpa lot to take a different viewpoint and consider the inverse problemas a system of model and observation, with the state of the system(in the above EIT example this would be the potential of theelectric field inside the body) and the searched for parameter (theconductivity distribution in EIT) as unknowns.
Reconstructionmethods based on such a kind of formulation are often calledall-at-once methods since they consider the model and the observationsimultaneously, instead of trying to eliminate the state from thesystem, as it is done in the above mentioned forward operator basedmethods.
Inthis project we intend to further develop and advance themathematical theory for such all-at-once methods and widen theirrange of applicability. In particular we plan to generalize thementioned model-plus-observation-equation approach to formulationsbased on optimization problems rather than systems of equations.