Title: Robust finite element solvers for distributed hyperbolic optimal control problems Show Chinese title

Title: 分布式双曲最优控制问题的鲁棒有限元求解器

Abstract: 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. Show Chinese abstract

Abstract: 我们提出、分析并测试了新的鲁棒迭代求解器，用于求解由减少的最优性系统空间-时间有限元离散化产生的线性代数方程组，这些最优性系统定义了具有标准$L^2$和更一般的能量正则化的双曲分布式跟踪型最优控制问题的近似解。 与通常的时间步进方法不同，我们通过在完全非结构化单形网格上定义的空间-时间连续分段线性有限元基函数对最优性系统进行离散化。 如果我们旨在通过计算的有限元状态 $y_{\varrho h}$ 来得到给定期望状态 $y_d$ 的渐近最优逼近，那么正则化参数 $\varrho$ 的最佳选择通过关系式 $\varrho=h^4$ 和 $\varrho=h^2$ 分别与时空有限元网格尺寸 $h$ 相关联，分别针对 $L^2$ 和能量正则化。 对于这种设置，我们可以为约简的有限元最优系统构建鲁棒的（并行）迭代求解器。 这些结果可以推广到适应网格尺寸局部行为的可变正则化参数，在自适应网格细化的情况下，网格尺寸可能会发生剧烈变化。 数值结果牢固地说明了理论发现。