Solving inverse problems without using forward operators

Inverse problems generally speaking determine causes for desired or observed effects. An example for this is the reconstruction of structures inside the human body from measurements outside, as it is done in medical imaging. In particular, in electrical impedance tomography one measures (via electrodes) the voltage pattern on the body surface corresponding to different imposed current patterns on the body surface. These patterns are crucially influenced by the conductivity distribution inside the body, so here the conductivity distribution is the cause for the observed voltage-current effects on the surface. Inverting this cause-to-effect map one can recover the conductivity distribution inside, which by assigning typical conductivity values for e.g., lungs, heart, benign and malignant tissue, etc., gives an image of the interior of the body. Inverse problems have many other applications ranging from the characterization of materials via the detection of defects inside devices to calibration of models in biology as well as economic and and social sciences. Computational methods for solving inverse problems usually rely on some kind of inversion of the mentioned cause-to-effect map, which is also called forward operator. However, this forward operator is often compuationally quite expensive to evaluate or might even not be well-defined. In such cases it can help a lot to take a different viewpoint and consider the inverse problem as a system of model and observation, with the state of the system (in the above EIT example this would be the potential of the electric field inside the body) and the searched for parameter (the conductivity distribution in EIT) as unknowns. Reconstruction methods based on such a kind of formulation are often called all-at-once methods since they consider the model and the observation simultaneously, instead of trying to eliminate the state from the system, as it is done in the above mentioned forward operator based methods. In this project we intend to further develop and advance the mathematical theory for such all-at- once methods and widen their range of applicability. In particular we plan to generalize the mentioned model-plus-observation-equation approach to formulations based on optimization problems rather than systems of equations.

Inverse problems generally speaking determine causes for desired or observed effects. An example for this is the reconstruction of structures inside the human body from measurements outside, as it is done in medical imaging. In particular, in electrical impedance tomography one measures (via electrodes) the voltage pattern on the body surface corresponding to different imposed current patterns on the body surface. These patterns are crucially influenced by the conductivity distribution inside the body, so here the conductivity distribution is the cause for the observed voltage-current effects on the surface. Inverting this cause-to-effect map one can recover the conductivity distribution inside, which by assigning typical conductivity values for e.g., lungs, heart, benign and malignant tissue, etc., gives an image of the interior of the body. Inverse problems have many other applications ranging from the characterization of materials via the detection of defects inside devices to calibration of models in biology as well as economic and and social sciences. Computational methods for solving inverse problems usually rely on some kind of inversion of the mentioned cause-to-effect map, which is also called forward operator. However, this forward operator is often compuationally quite expensive to evaluate or might even not be well-defined. In such cases it can help a lot to take a different viewpoint and consider the inverse problem as a system of model and observation, with the state of the system (in the above EIT example this would be the potential of the electric field inside the body) and the searched for parameter (the conductivity distribution in EIT) as unknowns. Reconstruction methods based on such a kind of formulation are often called all-at-once methods since they consider the model and the observation simultaneously, instead of trying to eliminate the state from the system, as it is done in the above mentioned forward operator based methods. In this project we intend to further develop and advance the mathematical theory for such all-at-once methods and widen their range of applicability. In particular we plan to generalize the mentioned model-plus-observation-equation approach to formulations based on optimization problems rather than systems of equations.