发表机构
George Mason University; Louisiana State University（乔治梅森大学; 路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种针对控制与状态约束稀疏最优控制的BDDC预条件子，在无限维层面构造，通过两级加性Schwarz实现，数值实验表明其迭代次数对网格细化近乎无关，且富化粗空间可提升收敛性。
AI 中文摘要
我们针对椭圆最优控制问题中由主动集半光滑牛顿线性化产生的界面系统，开发了一种基于约束的平衡域分解（BDDC）预条件子，该问题具有盒约束控制、$L^1$稀疏性和Moreau-Yosida正则化状态约束。域分解和BDDC构造直接在无限维层面进行。消除内部变量后，得到由局部子域算子组装的状态和伴随迹的Schur补。我们证明了全局-局部等价性结果，并在可验证的原始约束条件下建立了局部和部分组装问题的适定性。BDDC算子具有两级加性Schwarz表示，包含独立的局部求解和有限维粗求解。我们使用协调有限元实现该预条件子，并针对分布式和Neumann边界控制进行了测试。在子域直径$H$与网格尺寸$h$之比固定的情况下，GMRES迭代次数几乎与网格细化无关，且随着该比值的增加仅适度增长。丰富粗空间显著改善了收敛性，并降低了对控制正则化和稀疏性权重的敏感性。对于状态约束问题，将Moreau-Yosida惩罚参数按$h^2$比例缩放，在网格细化时GMRES迭代次数几乎保持恒定。
英文摘要
We develop a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the interface systems arising from active-set semi-smooth Newton linearizations of elliptic optimal control problems with box-constrained controls, $L^1$-sparsity, and Moreau--Yosida regularized state constraints. The domain decomposition and BDDC construction are formulated directly at the infinite-dimensional level. Eliminating the interior variables yields a Schur complement for the state and adjoint traces assembled from local subdomain operators. We prove a global--local equivalence result, establish well-posedness of the local and partially assembled problems under verifiable conditions on the primal constraints. The BDDC operator admits a two-level additive Schwarz representation with independent local solves and a finite-dimensional coarse solve. We implement the preconditioner using conforming finite elements and test it for distributed and Neumann boundary control. For a fixed ratio of subdomain diameter $H$ to mesh size $h$, the GMRES iteration counts are nearly independent of mesh refinement and grow only moderately as this ratio increases. Enriching the coarse space substantially improves convergence and reduces sensitivity to the control regularization and sparsity weights. For the state-constrained problem, scaling the Moreau--Yosida penalty parameter proportionally to $h^2$ yields nearly constant GMRES iteration counts as the mesh is refined.
