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

该项目的总体目标是对非光滑形状优化问题的理论和数值的数学严格方法。目标函数具有几何性质，即它们强调最佳形状的某些期望属性。在此过程中，我们将研究不同类别的函数，它们分别适用于表面平滑或几何分割任务。这些函数的非光滑性最终都是基于表面的法向量场，这是它们在每种情况下发挥作用的关键特征。上述考虑的主要动机是所谓的几何逆问题，其中从数据重构未知的几何形状。在非侵入性传感领域，例如夹杂物的检测，以及医学成像领域，都可以找到许多应用实例。我们正在研究的新颖几何函数允许对期望或期望的几何特性进行详细控制。在项目的第一期中，我们主要考虑法向量场的总表面变化，它具有保边的性质。另一方面，在第二阶段，我们首先研究可用于几何分割的函数，即根据某些特征对表面进行分类。例如，这些功能可以帮助表达对表面片段的特定方向的偏好。这使得引入晶体学、地质学和材料科学等领域的专业知识成为可能。此外，我们还考虑了基于广义二阶曲面法线总变分的泛函。因此，可以表示对表面的某些曲率特性的偏好。作为应用实例，在每种情况下，作为经典成像方式的电阻抗层析成像（EIT）问题都将与新的几何泛函结合成几何逆问题。与理论性质的研究同等重要的是一种有效而稳健的数值实现方法。为了实现这一目标，我们将开发一种ADMM方法，由于表面法线的固有特性，该方法必须结合微分几何工具。

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.