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

为了探索高维复杂工程应用的基本科学问题，如具有时变偏微分代数方程（PDAEs）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量应该几乎是线性增长的——这是最优速率。在这个联合项目中，我们希望结合现代解决策略来解决偏微分代数方程的时间依赖系统，如自适应多层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，包括非精确非单调sqp方法，通过内点或半光滑牛顿策略有效处理控制和状态约束。其中，优化方法以自适应的方式控制pdae求解器的不精确性和精度。基于后验误差估计的自适应使我们能够判断数值近似的质量，并使用模型确定适当的策略，以提高整体优化过程的准确性。成功的自适应方法大大节省了计算机时间和内存需求。它们可能意味着得到所考虑的优化问题的答案或不得到答案之间的差异。以辐射传热和弹性变形与传热之间的热-机械耦合为模型的玻璃冷却过程的最优边界控制问题，以及掺杂剂在硅中的再分布的最优控制，作为对状态和控制变量的限制必不可少的展示工程应用。

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.