Zhao-Zheng Liang
School of Mathematics and Statistics, Lanzhou University
P.R. China
We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equation. Due to the time-harmonic property of the control equation, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with a matrix of a particular two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degree of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2.
Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.