A new multigrid method for unconstrained parabolic optimal control problems

A second-order leapfrog finite difference scheme in time is proposed and developed for solving the first-order necessary optimality system of the distributed parabolic optimal control problems. Different from available approaches, the proposed leapfrog scheme for the two-point boundary optimality system is shown to be unconditionally stable and provides a second-order accuracy, though the classical leapfrog scheme usually is unstable. Moreover the proposed leapfrog scheme provides a feasible structure that leads to an effective implementation of a fast solver under the multigrid framework. A detailed mathematical proof for the stability of the proposed scheme is provided in terms of a new norm that is more suitable and stronger to characterize the convergence than the L2 norm often used in literature. Numerical experiments show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity.