Robust preconditioners for optimal control with time-periodic parabolic equation

Based on a new approximation of a Schur complement matrix and some preconditioning techniques considered in earlier papers, three robust preconditioners are presented for solving the KKT systems arising from discretizing optimal control problem with time-periodic parabolic equation. The eigenvalue bounds of the preconditioned matrices and some new theoretical results are obtained. Moreover, selection of quasi-optimal parameter is studied and an explicit expression of the quasi-optimal parameter is given. Numerical examples are given to illustrate the effectiveness of the three proposed optimized preconditioners.