We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the solution of elliptic tracking-type distributed optimal control problems with both the standard $L_2$ and the more general energy regularizations. If we aim at an approximation of the given desired state $y_d$ by the computed finite element state $y_h$ that asymptotically differs from $y_d$ in the order of the best $L_2$ approximation under acceptable costs for the control, then the optimal choice of the regularization parameter $\varrho$ is linked to the 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 efficient 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 case of adaptive mesh refinement. Similar results can be obtained for the space-time finite element discretization of the corresponding parabolic and hyperbolic optimal control problems.

翻译：本文提出、分析并测试了针对大规模线性代数方程组的新型迭代求解器，这些方程组源于约化最优性系统的有限元离散，该系统定义了椭圆跟踪型分布最优控制问题解的有限元近似，涉及标准$L_2$正则化及更一般的能量正则化。若我们期望通过计算得到的有限元状态$y_h$逼近给定期望状态$y_d$，且该逼近在控制代价可接受的前提下，与$y_d$的渐近误差达到最佳$L_2$逼近的阶数，则正则化参数$\varrho$的最优选择与网格尺寸$h$通过关系式$\varrho=h^4$（针对$L_2$正则化）和$\varrho=h^2$（针对能量正则化）相关联。在此设定下，我们可为约化有限元最优性系统构建高效的并行迭代求解器。这些结果可推广至随网格局部特征变化的自适应正则化参数——该参数在自适应网格细化情况下可能发生剧烈变化。类似结论还可拓展至相应抛物型与双曲型最优控制问题的时空有限元离散中。