Showing papers by "Zhong-Zhi Bai published in 2019"

Regularized HSS iteration methods for stabilized saddle-point problems

Abstract: We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval when the iteration parameter is close to and, furthermore, they can be clustered near and when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn-Hilliard image inpainting problem, as well as from the Gauss-Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

On banded M-splitting iteration methods for solving discretized spatial fractional diffusion equations

Abstract: For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M-splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition Also, we provide a multistep variant for the banded M-splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations Numerical experiments show the accuracy and efficiency of the multistep banded M-splitting iteration method

On multistep Rayleigh quotient iterations for Hermitian eigenvalue problems

Abstract: We present a multistep Rayleigh quotient iteration, as well as its inexact variant, for computing an eigenpair of a large sparse Hermitian matrix. Theoretical analysis shows that both exact and inexact multistep Rayleigh quotient iterations converge much faster than the exact and inexact Rayleigh quotient iterations, respectively. For the inexact multistep Rayleigh quotient iteration, we use the preconditioned conjugate gradient method to solve the inner linear systems, and find that significant saving in the number of inner iteration steps can be achieved when choosing a proper preconditioner. Numerical examples demonstrate effectiveness and superiority of our methods.

Computing eigenpairs of Hermitian matrices in perfect Krylov subspaces

Abstract: For computing the smallest eigenvalue and the corresponding eigenvector of a Hermitian matrix, by introducing a concept of perfect Krylov subspace, we propose a class of perfect Krylov subspace methods. For these methods, we prove their local, semilocal, and global convergence properties, and discuss their inexact implementations and preconditioning strategies. In addition, we use numerical experiments to demonstrate the convergence properties and exhibit the competitiveness of these methods with a few state-of-the art iteration methods such as Lanczos, rational Krylov sequence, and Jacobi-Davidson, when they are employed to solve large and sparse Hermitian eigenvalue problems.