Parameter-robust preconditioning for the optimal control of the wave equation
Parameter-robust preconditioning for the optimal control of the wave equation
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DOI：
10.1007/s11075-019-00720-y
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发表时间：
2019-05
影响因子：
2.1
通讯作者：
Jun Liu;J. Pearson
中科院分区：
文献类型：
--
作者：
Jun Liu;J. Pearson
本文提出并分析了一种新的匹配型Schur补预条件，用于求解表征波动方程最优控制的离散一阶必要最优性条件。与此相结合的是最近发展的二阶隐式有限差分格式，用于微分方程最优性系统的全时空离散化。导出了预条件系统的特征值界，从而深入了解了所应用的预条件Krylov子空间方法的收敛速度。数值例子验证了我们的理论分析，并证明了所提出的预调节器的有效性，特别是它对非常小的正则化参数和空间变量中所有网格尺寸的鲁棒性。
In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh sizes in the spatial variables.