Efficient Preconditioning for an Optimal Control Problem with the Time-Periodic Stokes Equations
Efficient Preconditioning for an Optimal Control Problem with the Time-Periodic Stokes Equations
复制标题
时间周期斯托克斯方程最优控制问题的有效预处理
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
W. Zulehner
中科院分区：
文献类型：
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作者：
Wolfgang Krendl;V. Simoncini;W. Zulehner
针对时间周期斯托克斯方程的最优控制问题，提出了一种实用的鲁棒预处理器。相应的最优系统的离散化导致线性系统具有鞍点形式的大型、稀疏且复杂的 4×4 块矩阵。我们提出了一种解耦策略，将系统简化为两个具有真正的 4×4 块矩阵的线性系统。基于 Krendl 等人针对时谐控制问题的预处理器的分析结果。 (Numer Math 124(1):183–213, 2013)，构造了一个实用的预处理器，它对于网格尺寸 h、频率 ω 和控制参数 ν 具有鲁棒性。结果通过预处理最小残差法的数值例子进行了说明。最后我们讨论替代停止标准。
For the optimal control problem with time-periodic Stokes equations a practical robust preconditioner is presented. The discretization of the corresponding optimality system leads to a linear system with a large, sparse and complex 4-by-4 block matrix in saddle point form. We present a decoupling strategy, which reduces the system to two linear systems with a real 4-by-4 block matrix. Based on analytic results on preconditioners for time-harmonic control problems in Krendl et al. (Numer Math 124(1):183–213, 2013), a practical preconditioner is constructed, which is robust with respect to the mesh size h, the frequency ω and the control parameter ν. The result is illustrated by numerical examples with the preconditioned minimal residual method. Finally we discuss alternative stopping criteria.