We consider standard tracking-type, distributed elliptic optimal control problems with $L^2$ regularization, and their finite element discretization. We are investigating the $L^2$ error between the finite element approximation $u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target) $\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size $h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble-Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.

翻译：我们考虑的是标准跟踪类型,分布式椭圆最佳控制问题($L $2),以及其有限的分解要素。我们正在调查国家美元和理想状态(目标)美元之间的限值元素近似 $úvarrho 美元和理想状态(目标)$br{u}美元之间的2美元差错,这导致最佳选择$\varrho = h ⁇ 4美元。我们发现,为了选择这种常规参数,我们可以设计简单的雅各比式的、具有先决条件的MINRES或Bramble-Pasciak CG 方法,使我们能够在算术操作和记忆需求方面以尽可能最佳的方式解决离散最佳的系统。理论结果得到若干基准问题的证实,其中涉及不同常规目标,包括不连续目标。