Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems
Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems
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DOI：
10.1137/130948045
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发表时间：
2016-03
期刊：
影响因子：
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通讯作者：
R. Herzog;Susann Mach
中科院分区：
文献类型：
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作者：
R. Herzog;Susann Mach
研究了具有逐点状态梯度约束的椭圆型最优控制问题。采用二次罚函数法和半光滑牛顿迭代法。提出了三种不同的预处理器，并分析了预处理后线性牛顿鞍点系统的谱特性与惩罚参数的关系。证明了最小正特征值的一个新的界。由于分析是在函数空间中进行的，因此它将确保合适的Krylov子空间方法（如Minres）在离散设置中的网格独立收敛行为。实现了一种带有预处理不精确牛顿解的路径跟踪策略，并提供了数值结果。
Elliptic optimal control problems with pointwise state gradient constraints are considered. A quadratic penalty approach is employed together with a semismooth Newton iteration. Three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear Newton saddle-point systems are analyzed dependent on the penalty parameter. A new bound for the smallest positive eigenvalue is proved. Since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable Krylov subspace methods such as Minres, also in discretized settings. A path-following strategy with a preconditioned inexact Newton solver is implemented and numerical results are provided.