解大型线性方程组的轮换重新开始Krylov子空间方法(英文)

Alternately restarted Krylov subspace methods for large linear systems of equations

重新开始Krylov子空间方法(包括Galerkin法和最小二乘法)是求解大型线性方程组的一类流行和重要的方法。然而,这类方法容易在收敛过程中发生中断或停滞现象。为了解决这一问题,本文提出一种新的重新开始格式,称之为轮换重新开始格式。该格式的基本思想是通过轮流使用方程组系数矩阵与其转置矩阵来生成Kry-lov子空间。轮换重新开始Krylov方法的迭代残量容易在各个特征向量方向上取得大致相等的收敛量,从而使得收敛得到改善。数值实验结果表明轮换重新开始Krylov子空间方法能够有效解决收敛失败的问题。

The restarted Krylov subspace methods, including the Galerkin method and the least-squares method, are popular and important for solving large linear systems of equations. However, the Galerkin method may suffer from serious breakdown, and the least-squares method may encounter complete stagnation. To overcome the problems, a new restarting scheme, called the alternately restarting scheme, is proposed in this paper. The underlying idea is to use the Krylov subspaces generated by the coefficient matrix and its transpose alternately. We show that for an alternately restarted Krylov method, its residual tends to get the same reduction in every eigenvector direction, and therefore its convergence can be significantly improved. Numerical experiments are conducted, which indicate that the alternately restarted Krylov subspace methods are efficient and robust.