We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, tracking-type optimal control problems with both the standard $L^2$ and the more general energy regularizations. In contrast to the usual time-stepping approach, we discretize the optimality system by space-time continuous piecewise-linear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state $y_d$ by the computed finite element state $y_{\varrho h}$, then the optimal choice of the regularization parameter $\varrho$ is linked to the space-time finite element mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements. The numerical results illustrate the theoretical findings firmly.

翻译：我们提出、分析并测试了新的鲁棒迭代求解器，用于求解由时空有限元离散化约化最优性系统生成的线性代数方程组，该系统定义了带有标准$L^2$正则化和更通用能量正则化的双曲型分布追踪最优控制问题的近似解。与常规的时间步进方法不同，我们采用定义在全非结构化单纯形网格上的时空连续分段线性有限元基函数对最优性系统进行离散化。若我们通过计算所得的有限元状态$y_{\varrho h}$追求对给定期望状态$y_d$的渐近最优逼近，则正则化参数$\varrho$的最优选择与时空有限元网格尺寸$h$相关联，分别满足$\varrho=h^4$（对应$L^2$正则化）和$\varrho=h^2$（对应能量正则化）。在此设定下，我们可为约化有限元最优性系统构造鲁棒的（并行）迭代求解器。这些结果可推广至随局部网格尺寸变化而调整的可变正则化参数——在自适应网格加密场景中，网格尺寸可能发生剧烈变化。数值结果有力地验证了理论发现。