Xi Yang

To further study the application of waveform relaxation methods in fluid dynamics in actual computation, this paper provides a general theoretical analysis of discrete-time waveform relaxation methods for solving linear DAEs. A class of discrete-time waveform relaxation methods, named discrete-time accelerated block successive overrelaxation (DABSOR) methods, is proposed for solving linear DAEs derived from discretizing time-dependent Stokes equations in space by using "Method of Lines". The analysis of convergence property and optimality of the DABSOR method are presented in detail. The theoretical results and the efficiency of the DABSOR method are verified by numerical experiments.

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, i.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iteration, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably good.