Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems

Abstract

We construct a preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration scheme for solving and preconditioning a class of block two-by-two linear systems arising from the Galerkin finite element discretizations of a class of distributed control problems. The convergence theory of this class of PMHSS iteration methods is established and the spectral properties of the PMHSSpreconditioned matrix are analysed. Numerical experiments show that the PMHSS preconditioners can be quite competitive when used to precondition Krylov subspace iteration methods such as GMRES.

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

“…The eigenvalue bounds for the preconditioned matrices given in Theorem 2.2 are valid, however, only when each of the matrices W and T is either positive or negative semidefinite and when null(W) ∩ null(T) = {0}. These observations equally apply to the special case 2 V = W of the PMHSS iteration methods introduced and discussed in [8,9], i.e., under these weaker conditions the PMHSS-preconditioned matrices F(α) −1 A are unitarily diagonalizable. For more recent discussions on this topic, we refer to [21] and the references therein.…”

“…being the symmetric and the skew-symmetric parts and A T being the transpose of the matrix A; see [1,14]. By modifying and preconditioning the HSS iteration method [10,11,13], recently Bai, Benzi, Chen and Wang [9] proposed and discussed a class of preconditioned modified HSS (PMHSS) iteration methods for solving the block two-by-two linear system (1.1); see also [7,8].…”

“…The PMHSS iteration method introduced in [9] naturally leads to a preconditioning matrix F(α) = (α + 1)P(α) αW + T 0 0 αW + T , with P(α) = 1 2α…”

“…Numerical results show that the exact and the inexact ABD-preconditioned MINRES or GMRES methods [19] lead to rapid convergence and outperform the exact and the inexact PMHSS-preconditioned GMRES methods, respectively; see [9].…”

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

“…Many efficient iterative methods have been studied in the literatures. For example, Uzawa-like methods ( [8,10,12]), SOR-like methods ( [7,14]), RPCG methods ( [6,9]), HSS-like methods ( [2][3][4][5]) and so on. We refer to [2] for algebraic properties for saddle point problem (1.7).…”