A Calculus for Non-Smooth Shape Optimization with Applications to Geometric Inverse Problems

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The overall goal of the project is a mathematically rigorous approach to the theory and numerics of non-smooth shape optimization problems. The objective functions are of a geometric nature, i.e., they emphasize certain desired properties of optimal shapes. In the process, we will examine different classes of functionals, which are, as they stand, suitable for surface smoothing or geometry segmentation tasks, respectively. The non-smoothness of these functionals, which ultimately are all based on the normal vector field of the surface, is a crucial feature in each case for their functioning.The main motivation for the above considerations are so-called geometric inverse problems, in which an unknown geometry is to be reconstructed from data. Numerous examples of applications for this can be found in the field of non-invasive sensing, for example for the detection of inclusions, but also in medical imaging. The novel geometric functionals we are examining allow a detailed control over expected or desired properties of the geometries to be identified. In the first phase of the project, we mainly considered the total surface variation of the normal vector field in this context, which has the property of being edge preserving.In the second phase, on the other hand, we first look at functionals that can be used for geometry segmentation, that is, the classification of a surface according to certain features. These functionals can, for instance, help to express a preference for specific orientations of the surface segments. This makes it possible to bring in expert knowledge in the field of crystallography, geology and material science, for example. Furthermore, we consider functionals based on the generalized, second-order total variation of the surface normals. Thereby, a preference for certain curvature properties of the surface can be expressed.As application examples, in each case problems of electrical impedance tomography (EIT) as a classical imaging modality are to be combined with the new geometric functionals into a geometric inverse problem. On an equal level with the investigation of the theoretical properties is always an efficient and robust numerical realization. To achieve this, we will develop an ADMM method, which will have to incorporate tools of differential geometry due to the intrinsic properties of the surface normal.