Model reduction by adaptive discretization in optimal control

该项目将采用目标导向自适应的概念来简化模型，以解决由偏微分方程（PDE）控制的最优控制问题。其基本框架是双加权残差法（DWR），该方法最初由R. Becker提出，并应用于有限元伽辽金法对偏微分方程进行自适应离散化。在这种方法中，对感兴趣的数量导出基于残差的加权后验误差估计，其中权重通过数值求解相关的对偶问题获得。由于使用了问题固有的灵敏度信息，这些后验误差估计适合于计算的特殊需要。这允许连续改进空间和时间离散化的控制，最终导致高度经济的离散化。在这个项目中，主要重点是非平稳最优控制问题，这对计算资源提出了特别高的要求，以及涉及控制和状态附加约束的问题。在这些情况下，自适应离散化的模型简化可能是最有用的。在pde约束最优控制的背景下，这些主题的研究只是最近才开始的，仍然有许多理论和实践上的开放性问题。

This project will employ the concept of goal-oriented adaptivity for model reduction in solving optimal control problems governed by partial differential equations (PDE). The underlying framework is the Dual Weighted Residual (DWR) method which was originally developed by R. Becker and the applicant for the adaptive discretization of PDE by the finite element Galerkin method. In this approach residual-based weighted a posteriori error estimates are derived for quantities of interest, where the weights are obtained by numerically solving an associated dual problem . Due to the use of problem inherent sensitivity information, these a posteriori error estimates are tailored to the special needs of the computation. This allows for successively improved control of spatial and time discretization which eventually results in highly economical discretization. In this project the main emphasis is on nonstationary optimal control problems, which pose particularly high requirements on computational resources, and on problems involving additional constraints for controls and states. In these cases model reduction by adaptive discretization may prove most useful. Research on these topics in the context of PDE-constrained optimal control has started only recently and there are still many theoretical as well as practical open questions.