We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton–Picard approach which divides the variable space into slow and fast modes. The division is performed either classically with eigenspace methods or with a novel two-grid approach. We prove mesh-independent convergence for the classical Newton–Picard preconditioner, present a complexity analysis, and show numerical results for the classical and the two-grid preconditioners. Moreover, the preconditioners compare favorably with existing symmetric positive definite Schur complement preconditioners in a Krylov method context.

## Citation

SIAM J. Sci. Comput., 34(2), A1214–A1239. (26 pages) doi: 10.1137/100807776