On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems

Abstract

Summary The Chebyshev accelerated preconditioned modified Hermitian and skew‐Hermitian splitting (CAPMHSS) iteration method is presented for solving the linear systems of equations, which have two‐by‐two block coefficient matrices. We derive an iteration error bound to show that the new method is convergent as long as the eigenvalue bounds are not underestimated. Even when the spectral information is lacking, the CAPMHSS iteration method could be considered as an exponentially converging iterative scheme for certain choices of the method parameters. In this case, the convergence rate is independent of the parameters. Besides, the linear subsystems in each iteration can be solved inexactly, which leads to the inexact CAPMHSS iteration method. The iteration error bound of the inexact method is derived also. We discuss in detail the implementation of CAPMHSS for solving two models arising from the Galerkin finite‐element discretizations of distributed control problems and complex symmetric linear systems. The numerical results show the robustness and the efficiency of the new methods.

Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

“…Other well-known preconditioning strategies for general two-by-two block matrices, such as block-triangular preconditioners, are also applicable, cf., e.g. [24,55,56]. We do not discuss them here any further.…”

PMHSS iteration method and preconditioners for Stokes control PDE-constrained optimization problems

“…The PMHSS preconditioner has nearly the same workload as the PRESB preconditioner. However, it can be used in the short-term recurrence iteration methods (e.g., minimal residual method (MIN-RES) and Chebyshev semi-iteration method) [5,18,19]. The convergence of the PMHSS iteration method is analyzed when W, T ∈ R n×n are symmetric positive semidefinite matrices.…”