Structure exploiting Galerkin schemes for optimization problems with pde constraints

利用伽辽金方案解决具有偏微分方程约束的优化问题的结构

• 批准号: 25269171
• 负责人: Professor Dr. Klaus Deckelnick
• 金额: --
• 依托单位: Institut für Analysis und Numerik (IAN)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/25269171?language=en
• 关键词: Structure; exploiting; Galerkin; schemes; optimization

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.

该项目与PDE约束优化问题（包括控制和状态约束）的量身定制离散概念和数值算法有关。与PDE约束的优化问题的数学分析和数值治疗必不可少，并在算法，分析和离散化中改善现有的新数学概念的发展。 PDE约束优化的主要目标是开发离散的概念和算法，这些概念和算法遵守优化的关系>恒定模拟的关系和现代大小的常数。为了在该项目中实现此目标，我们（a）提出了一个量身定制的离散概念，以通过包括控制约束在内的非线性PDE进行优化问题，以及（b）在PDE中使用状态约束中的PDE约束优化开发新的离散概念。对于这两种情况，我们都提供数值分析，包括收敛证明和适应性的数值算法。该关键思想在于，在离散级别上构成了无限二维KKT（Karush-kuhn-tucker）系统的尽可能多的结构，并适当地模仿KKT系统的功能分析关系。在第二个申请期内，我们将有能力将开发的离散策略与层次解决方案概念相结合，以解决PDE受限的优化问题，例如Multigrid方法，并将其纳入PDE受限优化策略的自适应改进策略。