In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set $\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup \left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed {parallel-in-time} preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around $\pm 1$, which leads to rapid convergence. When these parallel-in-time preconditioners are not fast diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners and the related advantages with respect to the relevant literature.

翻译：本文提出了一种新颖的预条件化Krylov子空间方法，用于求解波动方程的最优控制问题，其核心思想是显式识别该最优控制问题中线性系数矩阵序列的渐近谱分布。具体而言，我们首先证明波动控制问题所导出的全局求解系统与一个具有特征值分布的结构化系数矩阵序列相关联。随后，基于该谱分布及显式识别的符号函数，我们针对由两个块Toeplitz矩阵构成的鞍点系统，分别设计了一个理想预条件子和两个并行时间预条件子。对于理想预条件子，我们证明了预条件化后矩阵序列的所有特征值均属于集合$\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup \left(\frac{1}{2},\frac{3}{2}\right)$，且与零分离，从而确保采用最小残量法时收敛速度与网格无关。所提出的并行时间预条件子可通过快速傅里叶变换或离散正弦变换高效实现，其有效性在理论上得以证明，即预条件化后矩阵序列的特征值聚集于$\pm 1$附近，进而实现快速收敛。当这些并行时间预条件子无法快速对角化时，我们进一步提出了可高效求逆的改进版本。最后，通过若干数值算例验证了所推导的局部化与谱分布结果，并支持所提预条件子的有效性及其相较于相关文献的显著优势。