In this study, a novel preconditioner based on the absolute-value block $\alpha$-circulant matrix approximation is developed, specifically designed for nonsymmetric dense block lower triangular Toeplitz (BLTT) systems that emerge from the numerical discretization of evolutionary equations. Our preconditioner is constructed by taking an absolute-value of a block $\alpha$-circulant matrix approximation to the BLTT matrix. To apply our preconditioner, the original BLTT linear system is converted into a symmetric form by applying a time-reversing permutation transformation. Then, with our preconditioner, the preconditioned minimal residual method (MINRES) solver is employed to solve the symmetrized linear system. With properly chosen $\alpha$, the eigenvalues of the preconditioned matrix are proven to be clustered around $\pm1$ without any significant outliers. With the clustered spectrum, we show that the preconditioned MINRES solver for the preconditioned system has a convergence rate independent of system size. To the best of our knowledge, this is the first preconditioned MINRES method with size-independent convergence rate for the dense BLTT system. The efficacy of the proposed preconditioner is corroborated by our numerical experiments, which reveal that it attains optimal convergence.

翻译：本研究提出了一种基于绝对值块$\alpha$-循环矩阵近似的新型预处理器，专门针对演化方程数值离散中出现的非对称稠密块下三角Toeplitz（BLTT）系统而设计。该预处理器通过对BLTT矩阵的块$\alpha$-循环矩阵近似取绝对值构造而成。为应用该预处理器，原始BLTT线性系统通过时间反向置换变换转化为对称形式。随后，利用所提预处理器，采用预处理最小残差法（MINRES）求解器求解对称化后的线性系统。通过恰当选取$\alpha$，证明预处理后矩阵的特征值聚集在$\pm1$附近且无显著离群点。基于聚集谱特性，我们证明了预处理系统的最小残差法求解器的收敛速度与系统规模无关。据我们所知，这是首个针对稠密BLTT系统具有规模无关收敛速度的预处理MINRES方法。数值实验验证了所提预处理器的有效性，结果表明其达到了最优收敛性。