arXiv subjects

Zeyu Liao

Publications and source records attributed to Zeyu Liao.

2 recordsLinked to original sources

arXiv2025

Superlinear Convergence of GMRES for clustered eigenvalues and its application to least squares problems

The objective of this paper is to understand the superlinear convergence behavior of the GMRES method when the coefficient matrix has clustered eigenvalues. In order to understand the phenomenon, we analyze the convergence using the Vandermonde matrix which is defined using the eigenvalues of the coefficient matrix. Although eigenvalues alone cannot explain the convergence, they may provide an upper bound of the residual, together with the right hand side vector and the eigenvectors of the coefficient matrix. We show that when the coefficient matrix is diagonalizable, if the eigenvalues of the coefficient matrix are clustered, the upper bound of the convergence curve shows superlinear convergence, when the norm of the matrix obtained by decomposing the right hand side vector into the eigenvector components is not so large. We apply the analysis to explain the convergence of inner-iteration preconditioned GMRES for least squares problems.

math.NA↗

arXiv2022

A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.

math.NA↗