A new multigrid method for unconstrained parabolic optimal control problems

Abstract

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 L-2 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.