Showing papers by "Zhong-Zhi Bai published in 2018"
TL;DR: For solving large-scale systems of linear equations by iteration methods, an effective probability criterion for selecting the working rows from the coefficient matrix is introduced.
Abstract: For solving large-scale systems of linear equations by iteration methods, we introduce an effective probability criterion for selecting the working rows from the coefficient matrix and construct a ...
154 citations
TL;DR: The convergence of these methods when the linear system is consistent is proved, and it is shown that these methods can be more efficient than the greedy randomized Kaczmarz method if the relaxation parameter is chosen appropriately.
90 citations
TL;DR: An exact closed-form formula is conducted for the mean squared error of the iterate generated by the randomized Kaczmarz method, which completes the existing closed- form formula derived only for the tall coefficient matrix.
51 citations
19 citations
TL;DR: A class of respectively scaled Hermitian and skew-Hermitian splitting iteration method that can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems.
Abstract: Summary
For the discrete linear systems resulted from the discretization of the one-dimensional anisotropic spatial fractional diffusion equations of variable coefficients with the shifted finite-difference formulas of the Grunwald–Letnikov type, we propose a class of respectively scaled Hermitian and skew-Hermitian splitting iteration method and establish its asymptotic convergence theory. The corresponding induced matrix splitting preconditioner, through further replacements of the involved Toeplitz matrices with certain circulant matrices, leads to an economic variant that can be executed by fast Fourier transforms. Both theoretical analysis and numerical implementations show that this fast respectively scaled Hermitian and skew-Hermitian splitting preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems.
13 citations
TL;DR: The result shows that these eigenvalues are clustered around 0 +, 2 − , and a few points located on the unit circle centered at 1, as the iteration parameter is close to 0.
13 citations