Structure exploiting Galerkin schemes for optimization problems with pde constraints

该项目关注的是为pde约束优化问题（包括控制和状态约束）量身定制的离散概念和数值算法的发展。带偏微分约束的优化问题的数学分析和数值处理需要在算法、分析和离散化方面改进现有的数学概念和发展新的数学概念。pde约束优化的主要目标是发展离散的概念和算法，这些概念和算法符合优化的努力、模拟的努力和中等大小的常数之间的关系。为了在本项目中实现这一目标，我们(a)提出了一个针对包含控制约束的非线性偏微分方程优化问题的定制离散概念，以及(b)在具有状态约束的偏微分方程约束优化中开发了一个新的离散概念。对于这两种情况，我们提供了数值分析，包括收敛证明和适应的数值算法。关键思想在于在离散水平上尽可能多地保留无限维KKT （Karush-Kuhn-Tucker）系统的结构，并通过适当选择Ansätze所涉及的变量来适当地模拟KKT系统的泛函分析关系。在第二个应用阶段，我们将能够将开发的离散化策略与pde约束优化问题的分层解决概念（如多网格方法）相结合，并将其纳入pde约束优化策略的自适应细化策略中。

This project is concerned with the development of tailored discrete concepts and numerical algorithms for pde constrained optimization problems including control and state constraints. The mathematical analysis and numerical treatment of optimization problems with pde constraints necessitates the improvement of existing and the development of new mathematical concepts in algorithms, analysis and discretization. The major goal in pde constraint optimization consists in developing discrete concepts and algorithms which obey the relationeffort of optimization > constanteffort of simulationwith a constant of moderate size. In order to achieve this goal in this project we(a) propose a tailored discrete concept for optimization problems with nonlinear pdes including control constraints, and(b) develop a new discrete concept in pde constrained optimization with state constraints. For both cases we provide numerical analysis, including convergence proofs and adapted numerical algorithms.The key idea consists in conserving as much as possible structure of the infinite-dimensional KKT (Karush-Kuhn-Tucker) system on the discrete level, and to appropriately mimic the functional analytic relations of the KKT system through suitably chosen Ansätze for the variables involved. In a second application period we would be in position to combine the developed discretization strategies with hierarchical solution concepts for pde constrained optimization problems, such as multigrid methods, and to incorporate them into adaptive refinement strategies for pde constrained optimization strategies.