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

非光滑形状优化微积分及其在几何反问题中的应用

• 批准号: 314150341
• 负责人: Professor Dr. Roland Herzog
• 金额: --
• 依托单位: Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314150341?language=en
• 关键词: Calculus; Non; Smooth; Shape; Optimization

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.

该项目的总体目标是对非平滑形状优化问题的理论和数字进行数学上严格的方法。目标函数具有几何性质，即它们强调最佳形状的某些所需特性。在此过程中，我们将检查不同类别的功能，这些功能分别适用于表面平滑或几何分割任务。这些功能最终基于表面的正常矢量场的非平滑​​度，在每种情况下都是至关重要的。可以在非侵入性传感的领域中找到许多应用程序的例子，例如用于检测夹杂物，也可以在医学成像中找到。我们正在研究的新型几何功能允许确定对几何形状的预期或所需特性的详细控制。在项目的第一阶段中，我们主要考虑在这种情况下正常矢量场的总表面变化，该属性具有边缘保留的特性。在第二阶段，我们首先要查看可用于几何分割的功能，即，根据某些特征的表面分类。例如，这些功能可以帮助表达对表面段的特定方向的偏爱。例如，这使得在晶体学，地质和材料科学领域引入专家知识成为可能。此外，我们考虑了基于表面正态的广义，二阶总变化的功能。因此，可以表达对表面某些曲率特性的偏好。作为应用示例，在每种情况下，电阻抗断层扫描（EIT）的问题是经典成像方式，都应将新几何功能与几何几何函数结合在一起。在理论特性的研究中，同等水平始终是一种有效且坚固的数值实现。为了实现这一目标，我们将开发一种ADMM方法，该方法将由于表面正常的固有特性而必须合并差分几何的工具。