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光滑形状优化问题的理论和数值提供严格的数学方法。目标函数具有几何性质，即它们强调最佳形状的某些所需属性。在此过程中，我们将研究不同类别的泛函，就其本身而言，它们分别适用于表面平滑或几何分割任务。这些泛函的非光滑性最终都基于表面的法向量场，在每种情况下都是其功能的关键特征。上述考虑的主要动机是所谓的几何逆问题，其中未知的几何形状将从数据中重建。在非侵入式传感领域可以找到许多这样的应用示例，例如用于检测夹杂物，而且还可以在医学成像领域找到。我们正在研究的新颖几何泛函允许对要识别的几何形状的预期或期望属性进行详细控制。在项目的第一阶段，我们主要考虑了在这种情况下法向量场的总表面变化，它具有边缘保持的特性。另一方面，在第二阶段，我们首先考虑可以用于几何分割，即根据某些特征对表面进行分类。例如，这些泛函可以帮助表达对表面片段特定方向的偏好。这使得引进晶体学、地质学和材料科学等领域的专业知识成为可能。此外，我们考虑基于表面法线的广义二阶全变分的泛函。由此，可以表达对表面的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）作为经典成像模态的问题将与新的几何泛函组合成几何反问题。与理论性质的研究处于同等水平的始终是高效且稳健的数值实现。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。