In this work, we propose an absolute value block $\alpha$-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block $\alpha$-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block $\alpha$-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block $\alpha$-circulant preconditioner, which fills the gap between block $\alpha$-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen $\alpha$ the MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

翻译：本文提出了一种绝对值块 $\alpha$-循环预处理子，用于最小残差法（MINRES）求解波动方程离散化产生的全系统。由于近期被证明有效的原始块 $\alpha$-循环预处理子通常是非厄米的，因此不能直接用作 MINRES 的预处理子。受文献 [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018] 中提出的绝对值块循环预处理子的启发，我们提出了块 $\alpha$-循环预处理子的绝对值版本。提出的预处理子是首个厄米正定变体的块 $\alpha$-循环预处理子，填补了块 $\alpha$-循环预处理与预处理 MINRES 求解器领域之间的空白。预处理子的矩阵-向量乘法可通过快速傅里叶变换高效实现。理论上，我们证明了对于适当选取的 $\alpha$，采用所提预处理子的 MINRES 求解器具有与矩阵规模无关的线性收敛速率。据我们所知，这是首次在理论和性能两方面推广原始绝对值块循环预处理子的尝试。数值实验验证了预处理子的有效性，表明可以达到预期的最优收敛性。