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

The preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method and the corresponding preconditioning technique can achieve satisfactory results for solving optimal control problems governed by Poisson’s equation. We explore the feasibility of such a method and preconditioner for solving optimization problems constrained by the more complicated Stokes system. Theoretical results demonstrate that the PMHSS iteration method is convergent because the spectral radius of the iterative matrix is less than 22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {\sqrt {2}}{2}$\end{document}. Additionally, the PMHSS preconditioner still clusters eigenvalues on a unitary segment. It guarantees that the convergence of the PMHSS iteration method and preconditioning is independent of not only discretizing mesh size, but also of the Tikhonov regularization parameter. A more effective preconditioner is proposed based on the PMHSS preconditioner. The proposed preconditioner avoids the inner iterations when solving saddle point systems appearing in the generalized residual equations. Furthermore, it is still convergent and maintains its independence of parameter and mesh size.