波动方程all-at-once系统的快速α循环绝对值预处理

A Fastα-circulant Absolute Value Preconditioner for All-at-once Systems from Wave Equations

为了加快预处理MINRES方法求解波动方程all-at-once系统的收敛速度,基于绝对值预处理子和块状三对角Toeplitz预处理子,提出一种新的α循环绝对值预处理子。理论上证明了预处理矩阵可近似分裂成正交矩阵与低秩矩阵的和,且其特征值聚集在±1附近,保证了预处理MINRES方法的快速收敛性质。数值实验结果进一步表明了新预处理子的有效性。

In order to accelerate convergence rate of the preconditioned MINRES method for all-atonce systems from wave equations,we propose a newα-circulant absolute value preconditioner based on the absolute value preconditioners and the block tridiagonal Toeplitz preconditioners.Furthermore,we prove that the corresponding preconditioned matrix can be approximately split into the sum of the orthogonal matrix and the low-rank matrix,and its eigenvalues are clustered around±1,which leads to fast convergence rate of the preconditioned MINRES method.Numerical results also demonstrate the effectiveness of the new preconditioner.