Semi-Smooth Newton Methods on Shape Spaces

形状空间上的半光滑牛顿法

• 批准号: 423771068
• 负责人: Professor Dr. Volker Schulz
• 金额: --
• 依托单位: Institut für Numerische Mathematik und Optimierung
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423771068?language=en
• 关键词: Semi; Smooth; Newton; Methods; Shape

The aim of this proposal is to set up a novel approach for investigating analytically and solving computationally shape optimization problems constrained by variational inequalities (VI) in shape spaces. In contrast to classical VIs, where no explicit dependence on the domain is given, VI constrained shape optimization problems are in particular highly challenging because of two main reasons: Firstly, one needs to operate in inherently non-linear, non-convex and infinite-dimensional shape spaces. Secondly, one cannot expect for an arbitrary shape functional depending on solutions to VIs the existence of the shape derivative or to obtain the shape derivative as a linear mapping, which imply that the adjoint state cannot be introduced and, thus, the problem cannot be solved directly without any regularization techniques. Within project P20 'Optimizing variational inequalities on shape manifolds' in an SPP, theoretical results on volumetric shape derivatives for shape optimization problems constrained by the obstacle problem are provided and an efficient optimization algorithm is formulated. This proposal aims at extending the approaches established within another project to general, in the classical sense non-shape differentiable VI constrained problems. The main idea of this proposal is to consider Newton-shape derivatives instead of classical shape derivatives in order to formulate first-order necessary optimality conditions. Setting up a Newton-shape derivative scheme is the guiding principle for the analytical and numerical investigations within this project. More precisely, the resulting scheme enables the analytical and computational treatment of shape optimization problems constrained by VIs which are non-shape differentiable in the classical sense such that these can handled and solved without any regularization techniques leading often only to approximated shape solutions. Moreover, such a scheme opens the door for formulating higher order optimization methods like semi-smooth Newton methods on shapes spaces. Besides setting up a Newton shape derivative scheme, further goals of this project are investigations in the area of shape optimization for VIs regarding appropriate shape space formulations, existence and well-posedness of solutions including stationary concepts in shape spaces, semi-smooth Newton methods on shape spaces, mesh independent algorithmic approaches, robust treatment of uncertainties and solution approaches to application problems like, e.g. from the field of (thermo-)mechanics. Besides that, the shape space approach together with its novel Newton shape derivative scheme provides a basis for cooperation with other projects addressing shape based problem formulations.

本文的目的是建立一种新的方法来研究和计算形状空间中由变分不等式(VI)约束的形状优化问题。与传统的虚拟仪器相比，虚拟仪器的约束形状优化问题具有很高的挑战性，主要原因有两个：首先，人们需要在固有的非线性、非凸和无限维形状空间中进行操作。其次，人们不能期望任意形状泛函依赖于形状导数的解，或者将形状导数作为线性映射来获得，这意味着不能引入伴随状态，因此，如果没有任何正则化技术，问题就不能直接求解。在P20项目“优化形状流形上的变分不等式”中，给出了障碍约束形状优化问题的体积形状导数的理论结果，并给出了一个有效的优化算法。这一建议旨在将另一个项目中建立的方法推广到一般的、经典意义上的不可微VI约束问题。这一建议的主要思想是考虑牛顿型导数而不是经典的形状导数，以建立一阶必要条件。建立牛顿型导数格式是本项目分析和数值研究的指导原则。更准确地说，所得到的方案使得能够分析和计算由VIS约束的形状优化问题，这些问题在经典意义上是不可微的，使得这些问题可以在没有任何正则化技术的情况下被处理和解决，该正则化技术通常只导致近似的形状解。此外，这样的方案为在形状空间上建立像半光滑牛顿法这样的高阶优化方法打开了大门。除了建立牛顿形状导数格式外，本项目的另一个目标是研究可视化形状优化领域中适当的形状空间公式、解的存在性和适定性，包括形状空间中的静止概念、形状空间上的半光滑牛顿方法、网格无关的算法方法、不确定性的稳健处理以及应用问题的求解方法，例如从(热)力学领域。此外，形状空间方法及其新颖的牛顿形状导数格式为与其他解决基于形状的问题的公式的项目合作提供了基础。