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.

本计画主要针对包含控制与状态限制之偏微分方程限制最佳化问题，发展特制离散化概念与数值演算法。对偏微分方程约束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，并发展新的数学概念。偏微分方程约束优化的主要目标在于发展离散的概念和算法，使其服从优化的关系 >恒定的模拟效果，具有中等大小的常数。为了达到这个目标，在这个计画中，我们（a）提出一个适用于包含控制限制的非线性偏微分方程最佳化问题的离散化概念，以及（B）在包含状态限制的偏微分方程最佳化问题中发展一个新的离散化概念。对于这两种情况，我们提供了数值分析，包括收敛性证明和适应的数值algorithm.The关键思想在于尽可能多地保存的无限维KKT（Karush-Kuhn-Tucker）系统的离散水平上的结构，并适当地模仿KKT系统的功能分析关系，通过适当选择Ansätze的变量。在第二个应用阶段，我们将能够将联合收割机与用于偏微分方程约束优化问题的分层求解概念相结合，如多重网格方法，并将其纳入偏微分方程约束优化策略的自适应精化策略中。