Adaptive multilevel SQP-methods for PDAE-constrained optimization with restrictions on control and state. Theory and Applications

用于具有控制和状态限制的 PDAE 约束优化的自适应多级 SQP 方法。

• 批准号: 25227538
• 负责人: Professor Dr. Jens Lang
• 金额: --
• 依托单位: Arbeitsgruppe Nichtlineare Optimierung
• 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/25227538?language=en
• 关键词: Adaptive; multilevel; SQP; methods; PDAE

To explore the fundamental scientific issues of high dimensional complex engineering applications such as optimal control problems with time-dependent partial differential algebraic equations (PDAEs) scalable numerical algorithms are requested. This means that the work necessary to solve increasingly larger problems should grow all but linearly - the optimal rate. In this joint project we want to combine modern solution strategies to solve time-dependent systems of partial differential algebraic equations such as adaptive multilevel finite elements methods and error-controlled linearly implicit time integrators of higher order with state-of-the-art optimization techniques including inexact nonmonoton SQP-methods with an efficient handling of control and state constraints by interior-point or semismooth Newton strategies, where the optimization method controls the inexactness and accuracy of the PDAE-solver in an adaptive way. Adaptivity based on a posteriori error estimates enables us to judge the quality of the numerical approximations and used models to determine appropriate strategies to improve the accuracy of the overall optimization process. Successful adaptive methods lead to substantial savings in computer time and memory requirements. They can mean the difference between getting an answer or not to the optimization problem considered.An optimal boundary control problem of the cooling down process of glass modelled by radiative heat transfer and thermo-mechanical coupling between elastic deformation and heat transfer, and an optimal control of dopant s redistribution in silicon serve as showcase engineering applications where restrictions on state and control variables are essential.

为了探索高维复杂工程应用的基本科学问题，例如依赖时间的偏微分代数方程 (PDAE) 的最优控制问题，需要可扩展的数值算法。这意味着解决越来越大的问题所需的工作应该几乎呈线性增长——最佳速率。在这个联合项目中，我们希望结合现代求解策略来求解偏微分代数方程的时间相关系统，例如自适应多级有限元方法和高阶误差控制线性隐式时间积分器以及最先进的优化技术包括不精确的非单调 SQP 方法，通过内点或半平滑牛顿策略有效处理控制和状态约束，其中优化方法控制 PDAE 求解器的不精确性和准确性自适应方式。基于后验误差估计的自适应性使我们能够判断数值近似的质量，并使用模型来确定适当的策略以提高整体优化过程的准确性。成功的自适应方法可以大大节省计算机时间和内存需求。它们可能意味着所考虑的优化问题是否得到答案之间的差异。通过辐射传热和弹性变形与传热之间的热机械耦合建模的玻璃冷却过程的最优边界控制问题以及最优控制硅中掺杂剂的重新分布可以作为工程应用的展示，其中对状态和控制变量的限制至关重要。