In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

翻译：本文针对参数化偏微分方程控制的变边界最优控制问题，提出定制化的模型降阶方法。所谓变边界控制，是指边界控制作用于系统时存在特定参数变化。这种特殊形式的问题有望从模型降阶中获益。事实上，该模型的快速可靠仿真在地球物理学与能源工程等多个应用领域具有极其重要的实用价值。然而，变边界控制问题中状态变量与伴随变量的参数化行为极为复杂且多样化：以状态解为例，当边界控制参数发生变化时，可能表现出输运现象；同时，问题还丧失了仿射结构。众所周知，经典模型降阶技术在此类场景下既无法保证精度也难以实现高效性。为此，我们借鉴处理波动现象时所采用的降阶思路，提出两种定制化方法：几何重构法与局部本征正交分解法。几何重构法通过将优化系统转换至参考域求解，避免了超降阶处理带来的复杂性，而局部本征正交分解法则在几何重构法不可行的普适场景下，通过构建局部基函数提升降阶解的精度。基于几何复杂度递增的两种数值实验，我们对标准本征正交分解法与上述两种策略进行了系统比较。