Multiobjective Optimal Control of Partial Differential Equations Using Reduced-Order Modeling

在几乎所有的技术应用中，在开发和操作过程中都涉及多个标准。例如快速但节能的车辆和既轻又稳定的建筑。由此产生的多目标优化问题的目标是计算最优妥协集，即所谓的帕累托集。然后，决策者可以从这个集合中选择一个合适的解决方案。在控制应用中，作为对外部条件变化的反应，可以在不同妥协之间快速切换。帕累托集通常由无限多个折衷解组成，因此其数值逼近比标量优化问题的求解要昂贵得多。这可能很快导致过高的计算成本，特别是在底层系统的解决方案计算成本很高的情况下。例如，当系统由偏微分方程（PDE）描述时就是这种情况。在这种情况下，经常使用替代模型，它可以比传统的有限元数值近似更快地求解。在非光滑偏微分方程的情况下，降低计算成本尤为重要，因为这些问题的解决成本通常比光滑问题高得多。然而，代理模型给系统引入了近似误差，在分析和开发数值算法时都必须对其进行量化和考虑。对于非光滑问题，目前关于该主题的文献很少。本项目的目标是发展有效的数值方法来解决受某类非光滑偏微分方程约束的多目标优化问题。第一步，推导非光滑pde约束问题的最优性条件，并分析Pareto集的层次结构。在此基础上，将开发用于计算帕累托集的算法来解决这些问题。这些方法将用于优化问题与最大项，接触问题，和时间依赖的混合和切换系统。为了处理数值上的困难，降低阶的建模技术——如降低基、适当正交分解和最近基于Koopman算子的方法——将被扩展到非光滑环境。这就要求在收敛分析中考虑不精确性。最后，将与优先方案的其他成员合作，将这些算法应用于若干不同的问题。

In almost all technical applications, multiple criteria are of interest – both during development as well as operation. Examples are fast butenergy efficient vehicles and constructions that have to be light as well as stable. The goal in the resulting multiobjective optimizationproblems is the computation of the set of optimal compromises – the so-called Pareto set. A decision maker can then select an appropriatesolution from this set. In control applications, it is possible to quickly switch between different compromises as a reaction to changes in theexternal conditions. The Pareto set generally consists of infinitely many compromise solutions, its numerical approximation is thereforeconsiderably more expensive than the solution of scalar optimization problems. This can quickly result in prohibitively large computationalcost, particularly in situations where solutions to the underlying systems are computationally expensive. For instance, this is the casewhen the system is described by a partial differential equation (PDE).In this context, surrogate models are frequently used that can be solved significantly faster than classical numerical approximations bythe finite element method. In the case of non-smooth PDEs, reducing the computational cost is particularly important since these problemsare often significantly more expensive to solve than smooth problems. However, the surrogate models introduce an approximation error intothe system, which has to be quantified and considered both in the analysis and the development of numerical algorithms. For nonsmoothproblems, literature on this topic is currently scarce. The goal of this project is the development of efficient numerical methods to solve multiobjective optimization problems that are constrained by certain classes of non-smooth PDEs. In the first step, optimality conditions for the non-smooth PDE-constrained problems will be derived, and the (hierarchical) structure of the Pareto sets will be analyzed. Building on this, algorithms for the computation of Pareto sets will be developed for these problems. The methods will be used for the optimization of problems with max-terms, contact problems, and time dependent hybrid and switched systems. In order to handle the numerical effort, reduced order modeling techniques – such as Reduced Basis, Proper Orthogonal Decomposition and more recent approaches based on the Koopman operator will be extended to the non-smooth setting. This requires the consideration of inexactness in the convergence analysis. Finally, the algorithms will be applied to several different problem settings in cooperation with other members of the Priority Programme.