Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates
Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates
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DOI：
10.3934/mcrf.2022003
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发表时间：
2022
影响因子：
1.2
通讯作者：
Q. Tran;Harbir Antil;Hugo S Díaz
中科院分区：
文献类型：
--
作者：
Q. Tran;Harbir Antil;Hugo S Díaz
我们考虑由参数化稳态麦克斯韦系统和高斯定律控制的最优控制问题。参数通过介电常数、磁导率和电荷密度输入。此外，假设参数集是紧凑的。我们通过有限元方法对电场进行离散化，并使用变分离散化概念进行控制。我们提出了最优控制问题的简化基方法，并建立了降阶解与原始全维问题解的一致收敛，前提是快照参数样本在参数集中是密集的，并具有适当的参数可分离性规则。最后，我们根据状态和伴随残差建立降阶解的绝对后验误差估计器以及相应的成本函数。
We consider an optimal control problem governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept for the control. We present a reduced basis method for the optimal control problem and establish the uniform convergence of the reduced order solutions to that of the original full-dimensional problem provided that the snapshot parameter sample is dense in the parameter set, with an appropriate parameter separability rule. Finally, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functions in terms of the state and adjoint residuals.