SPP 1253: Optimisation with Partial Differential Equations

在工业、医疗和经济应用的背景下，求解受约束的分布参数系统（DPS）优化问题是最具挑战性的问题之一。特别是在飞机设计、化学工程中的“移动床”工艺、晶体生长等方面，前向模拟之后的优化变量变化已被证明是低效的。相反，优化的设计和拓扑结构和控制过程中涉及偏微分方程（PDE），解释为DPS，必须同时处理，使现代数学方法与PDE的优化是相互关联的自适应目标为导向的仿真工具。在适当的离散化结构之后，优化变量的数量通常在高达数百万的范围内变化。直到最近，计算能力的巨大进步才使解决这种规模的问题成为可能。然而，为了完成这一任务，它是至关重要的利用和进一步探索原型应用程序的具体数学结构，并开发新的数学方法，结构开发算法，模型简化，并行性，适应性的数值方案为相应的最优性系统的基础上，后验误差估计和优化与PDE涉及控制和状态约束。自动微分方法在处理优化设计、形状和拓扑问题以及时间相关问题所涉及的大量数据方面将变得非常重要。

Solving optimisation problems subject to constraints involving distributed parameter systems (DPS) is one of the most challenging problems in the context of industrial, medical and economical applications. In particular, in the design of aircraft, "moving bed" processes in chemical engineering, crystal growth etc. the forward simulation followed by the variation of the optimisation variables has proved to be inefficient. Instead, the optimisation of design and topology of structures and the control of processes involving partial differential equations (PDEs), interpreted as DPS, has to be treated simultaneously such that modern mathematical methods for optimisation with PDEs are interlinked with adaptive goal-oriented simulation tools. After proper structure respecting discretisation, the number of optimisation variables varies typically in the range of up to several millions. It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilise and further explore the specific mathematical structure of prototype applications and to develop new mathematical approaches concerning structure exploiting algorithms, model reduction, parallelisability, adaptivity of numerical schemes for the corresponding optimality systems based on a posteriori error estimates and the optimisation with PDEs involving control and state constraints. Methods of automatic differentiation (AD) will turn out to be very important in handling the massive data involved in problems of optimal design, shape and topology as well as in time-dependent problems.