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

使用降阶建模的偏微分方程的多目标最优控制

• 批准号: 314151124
• 负责人: Professor Dr. Michael Dellnitz
• 金额: --
• 依托单位: Arbeitsgruppe Numerische Optimierung
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314151124?language=en
• 关键词: Multiobjective; Optimal; Control; Partial; Differential

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.

在几乎所有的技术应用中，多个标准都是令人感兴趣的-无论是在开发过程中还是在操作过程中。例如，快速节能车辆和结构必须轻便且稳定。多目标优化问题的目标是计算最优折衷的集合--所谓的帕累托集合。决策者可以从中选择一个合适的解决方案。在控制应用中，可以快速在不同的妥协之间切换，以应对外部条件的变化。Pareto集一般由无穷多个折衷解组成，因此它的数值逼近比标量优化问题的解要昂贵得多。这可能很快导致巨大的计算成本，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，经常使用替代模型，这种模型的求解速度明显快于有限元法的经典数值逼近。在非光滑偏微分方程的情况下，降低计算成本是特别重要的，因为这些问题往往是显着更昂贵的解决比光滑的问题。然而，代理模型引入了一个近似误差到系统中，这必须被量化，并考虑在分析和数值算法的发展。对于nonsmoothproblems，关于这个主题的文献目前是稀缺的。这个项目的目标是开发有效的数值方法来解决受某些非光滑偏微分方程约束的多目标优化问题。在第一步中，将推导出非光滑偏微分方程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，将针对这些问题开发计算帕累托集的算法。该方法将用于优化问题的最大项，接触问题，时间相关的混合和切换系统。为了处理数值工作，降阶建模技术-如缩减基，适当的正交分解和最近的方法的基础上的库普曼算子将扩展到非光滑设置。这就要求在收敛性分析中考虑不精确性。最后，将与优先方案的其他成员合作，将算法应用于若干不同的问题设置。