We are concerned with robust iterative solution methods for solving the Stokes optimal control problems. Two efficient preconditioners are proposed for the discretized saddle point linear systems arising from the velocity tracking of the Stokes control problem. The proposed preconditioners are similar in structure to and can be viewed as modifications of the preconditioner in Axelsson et al. (2017) [2], which are economic to implement in an inner-outer framework within Krylov acceleration. They can lead to similar tight and problem independent eigenvalue distribution results for the preconditioned matrices, which yield rates of convergence independent of both the regularization parameter and refinement level. Moreover, we also give inexact variants of the proposed preconditioners, which avoid the inner-outer implementations utilizing preconditioned GMRES methods as inner loops. Numerical experiments indicate that the proposed preconditioners demonstrate robust performance and comparable to some existing preconditioners when used to accelerate the Krylov subspace methods.

我们关注用于解决斯托克斯最优控制问题的鲁棒迭代求解方法。针对离散斯托克斯点线性系统，针对斯托克斯控制问题的速度跟踪，提出了两个有效的预处理器。所提出的预处理器的结构与Axelsson等人的预处理器的修改类似。（2017）[2]，这是在Krylov加速过程中在内部-外部框架中实施的经济方法。对于预条件矩阵，它们可以导致相似的紧密和独立于特征的特征值分布结果，其收敛速度与正则化参数和细化级别无关。此外，我们还给出了拟议的预处理器的不精确变体，这避免了使用预处理的GMRES方法作为内部循环的内部-外部实现。数值实验表明，当用于加速Krylov子空间方法时，所提出的预处理器表现出鲁棒的性能，并且可以与某些现有的预处理器相媲美。