Fast solver of optimal control problems constrained by Ohta-Kawasaki equations

This paper is concerned with fast solver of distributed optimal control problems constrained by a nonlocal Cahn-Hilliard equation. By eliminating the control variable, a linear system on four-by-four block matrix form is obtained after discretization. Deforming the corresponding coefficient matrix into a form with special structure, an efficient preconditioner that can be utilized in an inner-outer way is designed, which leads to a fast Krylov subspace solver, that is robust with respect to mesh sizes, model parameters, and regularization parameters. Moreover, we prove that the eigenvalues of the corresponding preconditioned system are all real. Numerical experiments are presented to illustrate the robustness of the proposed solution methods.