In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block $\omega$-circulant based preconditioners for the all-at-once linear system arising from the concerned optimal control problem, where both first order and second order time discretization methods are considered. The proposed preconditioners can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix are clustered around $\pm 1$, which leads to rapid convergence when the minimal residual method is used. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners are justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.

翻译：本文针对抛物型方程的最优控制问题，提出了一类新颖的预处理Krylov子空间求解方法。具体而言，我们为所关注的最优控制问题中产生的全一次性线性系统，构建了一族基于块$\omega$-循环的预处理子，其中同时考虑了一阶和二阶时间离散方法。所提出的预处理子可以通过快速傅里叶变换以时间并行方式高效对角化，其有效性在理论上体现为预处理后矩阵的特征值聚集在$\pm 1$附近，这保证了在使用最小残差法时能获得快速收敛。当采用广义最小残差法时，预处理后矩阵的奇异值被证明聚集在1附近，从而验证了所提预处理子的有效性。数值结果进一步证明了我们提出的求解器的高效性。