Goal Oriented Mesh Adaptivity for Constrained Optimal Control and Optimization Problems

研究者建议发展目标导向网格自适应的概念，用于偏微分方程的约束最优控制和结构优化问题的数值解。对于纯仿真，即不进行优化的情况，基于高效可靠的后验误差估计的网格自适应是一种成熟的有限元离散偏微分方程的求解工具。残差类型的后验误差估计依赖于残差相对于问题的近似解的适当评估，并通过元素和面或边残差导致廉价可计算的局部误差项。误差通常在与底层函数空间相关的范数中估计。对偶方法涉及伴随问题，并允许人们推导出关于从全局范数到局部，甚至是点方向的各种误差函数的明显上界。其思想是将目标泛函视为给定微分方程伴随方程的右侧，从而提供原始误差与伴随方程误差之间的乘法关系。研究者将系统地研究目标导向的网格自适应控制和状态约束最优控制问题，以及结构优化问题，如形状和拓扑优化，在状态和设计变量上具有相等和不等式约束。特别是，他将考虑不同的目标量，包括目标函数和约束满足/违反，并研究它们对网格自适应和近似解精度的影响。在此基础上，他将开发、分析和实现面向目标的后验误差估计器。这将辅以广泛的数值研究，以证明所开发的工具在选定的最优控制和优化问题上的效率和可靠性。偏微分方程描述的系统的最优控制和结构优化对技术相关设备和系统的成本效益发展有着深远的影响。基于高效可靠的目标导向后验误差估计的自适应网格细化和粗化是数值设计研究的重要算法工具，有助于提高设备和系统的功能，而无需依赖于成本密集的原型生产。该项目将向研究生介绍最先进的优化和数值模拟方法。该材料将用于研究生和本科生课程。

The investigator proposes to develop concepts of goal oriented mesh adaptivity for the numerical solution of constrained optimal control and structural optimization problems for partial differential equations. In case of pure simulation, i.e., without optimization, mesh adaptivity on the basis of efficient and reliable a posteriori error estimators for finite element discretized partial differential equations is a well-established tool. Residual type a posteriori error estimators rely on the appropriate evaluation of the residual with respect to an approximate solution of the problem and lead to cheaply computable, local error terms by means of element and face or edge residuals. The error is typically estimated in norms associated with the underlying function space. The dual approach involves the adjoint problem and allows one to derive sharp upper bounds for the error with respect to various error functionals ranging from global norms to local, even pointwise, quantities. The idea is to consider the target functional as the right hand side in the adjoint of the given differential equation which provides a multiplicative relation between the error in the original and the adjoint equation. The investigator will systematically study goal oriented mesh adaptivity for control and state constrained optimal control problems and structural optimization problems such as shape and topology optimization with equality and inequality constraints on the state and design variables. In particular, he will consider different target quantities, including the objective functional and constraint satisfaction/violation, and investigate their impact on mesh adaptation and accuracy of the approximate solution. On this basis, he will develop, analyze and implement goal oriented a posteriori error estimators. This will be complemented by extensive numerical studies to document the efficiency and reliability of the developed tools for selected optimal control and optimization problems.The optimal control and structural optimization of systems described by partial differential equations has a deep impact on the cost effective development of technologically relevant devices and systems. Adaptive mesh refinement and coarsening on the basis of efficient and reliable goal oriented a posteriori error estimators is a significant algorithmic tool for numerical design studies which contribute to improve the functionality of the devices and systems without resorting to the cost intensive production of prototypes. The project will introduce graduate students to both state-of-the-art optimization and numerical simulation methods. The material will be used in graduate and undergraduate courses.