A new semi-smooth Newton multigrid method for parabolic PDE optimal control problems

A new semi-smooth Newton (SSN) multigrid algorithm is proposed for solving the discretized first order necessary optimality systems that characterize the optimal solutions of a class of 2D semi-linear parabolic PDE optimal control problems with control constraints. To achieve a second-order accurate finite difference discretization, we use a leapfrog scheme (with the second-order backward differentiation formula (BDF2)) in time and a standard 5-point stencil in space. The derived well-structured discretized Jacobian matrices greatly facilitate the development of effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the efficiency of the proposed method, which validates the second-order accuracy in solution approximations and the optimal linear complexity in computational time.