In this work, we propose a novel diagonalization-based preconditioner for the all-at-once linear system arising from the optimal control problem of parabolic equations. The proposed preconditioner is constructed based on an $\epsilon$-circulant modification to the rotated block diagonal (RBD) preconditioning technique, which can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion. \textcolor{black}{To our knowledge, this marks the first application of the $\epsilon$-circulant modification to RBD preconditioning. Before our work, the studies of PinT preconditioning techniques for the optimal control problem are mainly focused on $\epsilon$-circulant modification to Schur complement based preconditioners, which involves multiplication of forward and backward evolutionary processes and thus square the condition number. Compared with those Schur complement based preconditioning techniques in the literature, the advantage of the proposed $\epsilon$-circulant modified RBD preconditioning is that it does not involve the multiplication of forward and backward evolutionary processes. When the generalized minimal residual method is deployed on the preconditioned system, we prove that when choosing $\epsilon=\mathcal{O}(\sqrt{\tau})$ with $\tau$ being the temporal step-size , the convergence rate of the preconditioned GMRES solver is independent of the matrix size and the regularization parameter. Such restriction on $\epsilon$ is more relax than the assumptions on $\epsilon$ from other works related to $\epsilon$-circulant based preconditioning techniques for the optimal control problem. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.

翻译：本文针对抛物型方程最优控制问题产生的全一次性线性系统，提出了一种新颖的基于对角化的预处理器。该预处理器基于对旋转块对角预处理器技术的$\epsilon$-循环修正构造而成，可通过快速傅里叶变换以并行时间方式高效实现对角化。\textcolor{black}{据我们所知，这是$\epsilon$-循环修正技术在RBD预处理中的首次应用。在本文工作之前，针对最优控制问题的并行时间预处理技术研究主要集中于基于Schur补的预处理器的$\epsilon$-循环修正，这类方法涉及前向与后向演化过程的乘积运算，从而导致条件数平方级增长。与文献中基于Schur补的预处理技术相比，本文提出的$\epsilon$-循环修正RBD预处理技术的优势在于不涉及前向与后向演化过程的乘积运算。当在预处理系统上采用广义最小残差法时，我们证明：若选择$\epsilon=\mathcal{O}(\sqrt{\tau})$（其中$\tau$为时间步长），则预处理GMRES求解器的收敛速率与矩阵规模和正则化参数无关。该$\epsilon$取值条件相较于其他基于$\epsilon$-循环预处理技术的最优控制问题相关研究中的假设更为宽松。数值实验结果验证了所提出求解器的有效性。