A note on preconditioners for complex linear systems arising from PDE-constrained optimization problems

In this note, the block-diagonal preconditioner proposed and the block triangular proposed in Krendl et al. (2013) and Pearson and Wathen (2012), respectively, are further studied and optimized. Two improved preconditioners are proposed for solving a class of complex linear systems arising from optimal control with time-periodic parabolic equation. Theoretical analyses show that the eigenvalues of the improved block-diagonal and block triangular preconditioned matrices are located in [−1,−2/2)∪(2/2,1] and (1/2,1](1/2,1], respectively. Numerical experiments illustrate the feasibility and effectiveness of these preconditioners for Krylov subspace methods.