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：本研究开发了一种基于绝对值块$\alpha$-循环矩阵近似的新型预条件子，专门针对演化方程数值离散过程中产生的非对称稠密块下三角Toeplitz（BLTT）系统。该预条件子通过对BLTT矩阵的块$\alpha$-循环矩阵近似取绝对值构造而成。为应用此预条件子，原始BLTT线性系统通过时间反转置换变换转化为对称形式。随后，采用预条件极小残差法（MINRES）求解器对对称化线性系统进行求解。当$\alpha$选取适当时，预条件矩阵的特征值被证明聚集在$\pm1$附近，且无显著异常值。基于这种聚集谱，我们证明预条件系统的预条件MINRES求解器具有与系统规模无关的收敛速度。据我们所知，这是首个针对稠密BLTT系统实现规模无关收敛速度的预条件MINRES方法。数值实验验证了所提预条件子的有效性，表明其能达到最优收敛。