Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate properties significantly complicate every aspect of their analysis, from the derivation of first-order optimality conditions to their discrete approximation and the choice of a suitable solution algorithm. In this paper, we investigate a general class of minimization problems with TV-regularization, comprising both continuous and discretized control spaces, from a convex geometry perspective. This leads to a variety of novel theoretical insights on minimization problems with total variation regularization as well as tools for their practical realization. First, by studying the extremal points of the respective total variation unit balls, we enable their efficient solution by geometry exploiting algorithms, e.g. fully-corrective generalized conditional gradient methods. We give a detailed account on the practical realization of such a method for piecewise constant finite element approximations of the control on triangulations of the spatial domain. Second, in the same setting and for suitable sequences of uniformly refined meshes, it is shown that minimizers to discretized PDE-constrained optimal control problems approximate solutions to a continuous limit problem involving an anisotropic total variation reflecting the fine-scale geometry of the mesh.

翻译：总变差正则化在微分方程最优控制领域已被证明是一种有价值的工具。这主要归因于以下观察：TV惩罚项往往倾向于产生具有良好跳跃集的分段常数极小化子。然而，其复杂特性也显著增加了分析过程的每个环节的难度，从一阶最优性条件的推导到离散逼近及合适求解算法的选择均面临挑战。本文从凸几何视角研究一类具有TV正则化的广义极小化问题，涵盖连续与离散控制空间。该研究不仅为总变差正则化极小化问题提供了多种新颖的理论见解，还为其实际实现开发了实用工具。首先，通过研究相应总变差单位球的极值点，我们能够利用几何特征算法（例如完全校正广义条件梯度方法）高效求解这些问题。我们详细阐述了在空间域三角剖分上对控制变量采用分段常数有限元逼近时，此类方法的具体实现方案。其次，在同一框架下针对适当的一致加密网格序列，我们证明了离散PDE约束最优控制问题的极小化子能够逼近连续极限问题的解，该极限问题包含反映网格细观几何结构的各向异性总变差。