The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain-decomposition (DD) methods and reduced-order modelling (ROM). In particular, we consider an optimisation-based domain-decomposition algorithm for the parameter-dependent stationary incompressible Navier-Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal-control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward-facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain-decomposition algorithm.

翻译：本文旨在提出一种用于偏微分方程最优控制问题的模型降阶技术。我们结合了两种降低数学数值模型计算成本的方法：区域分解方法和降阶建模。具体而言，我们考虑了一种基于优化的区域分解算法，用于处理参数相关的稳态不可压缩Navier-Stokes方程。首先，问题在子域中通过界面耦合进行描述，并通过最优控制问题求解，从而在区域分解方法中实现子域问题的完全分离。在此基础上，针对所获得的最优控制问题构建了降阶模型；该过程基于本征正交分解技术和进一步的伽辽金投影。所提出的方法在两个流体动力学基准算例上进行了测试：稳态后向台阶流动和顶盖驱动方腔流动。数值测试表明，在问题维度和区域分解算法中的优化迭代次数方面，计算成本均得到了显著降低。