Model reduction by adaptive discretization in optimal control

这个项目将使用面向目标的自适应概念来解决由偏微分方程组(PDE)控制的最优控制问题的模型降阶。其基本框架是由R.Becker和应用有限元Galerkin方法自适应离散PDE的对偶加权残差(DWR)方法。在这种方法中，基于残差的加权后验误差估计是针对感兴趣的量，其中权重是通过数值求解相关的对偶问题来获得的。由于使用了问题固有的敏感性信息，这些后验误差估计是根据计算的特殊需要量身定做的。这允许连续改进对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，主要强调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的额外约束的问题。在这些情况下，通过自适应离散化进行模型简化可能被证明是最有用的。在偏微分方程约束下的最优控制的研究才刚刚起步，仍有许多理论和实践上的开放问题。

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.