Generalized nonlinear minimal residual (GNLMR) method for iterative algorithms

摘要

Most iterative methods for solving steady-state problems can be shown to be equivalent to solving time-dependent problems of either parabolic or hyperbolic type. The relaxation factor used in accelerating an iterative method to obtain the converged solution plays the same role as the time step size used in advancing the transient solution to the steady state solution for a time-dependent problem. With this transformation, one can expose the mechanism of the acceleration schemes. In the presented study, this time-dependent approach together with the single-iteration, multi-step algorithm are applied to generalize the nonlinear minimal residual (NLMR) method for iterative solutions of linear and nonlinear problems. Most importantly, both theoretical studies and numerical experiments confirm the monotone convergence behavior of the generalized NLMR method. With the multi-step algorithm, it is found that both the rate and the smoothness of convergence of the NLMR method can be improved even further. Several interesting problems that originated from this method are also discussed.