On Block-diagonal Preconditioned Accelerated Parameterized Inexact Uzawa Method For Saddle Point Problem

Large sparse linear systems of saddle point structure result from a wide variety of computation science and engineering applications, such as discrete approxima-tions of certain partial differential equations, interior point methods in constrained optimization, constrained and weighted least squares estimation, computational flu-id dynamics and so on. Recently, a large amount of work has been developed to solve these linear systems due to the challenges of its indefiniteness and poor spectral properties for solver developers.In this paper, by combining the accelerated parameterized inexact Uzawa (API-U) method with the block-diagonally preconditioned parameterized inexact Uzawa (PPIU) method for solving saddle point problems, we further generalize these two methods to the block-diagonally preconditioned accelerated parameterized inexac-t Uzawa (BDP-APIU) method and use it to solve both nonsingular and singular saddle point problems. Theoretical analyses show that the convergence and semi-convergence of this new method for solving nonsingular and singular saddle point problems can be guaranteed. In addition, the selections for quasi-optimal parame-ters of this method for solving both nonsingular and singular saddle point problems are discussed. Numerical examples are given to show the feasibility and effectiveness of the new method for solving both nonsingular and singular saddle point problems.