NEWTON-PICARD PRECONDITIONERS FOR TIME-PERIODIC PARABOLIC OPTIMAL CONTROL PROBLEMS

We prove existence and uniqueness of solutions of an optimization problem with time-periodic parabolic partial differential equation constraints and show that the solution inherits high smoothness properties from the given data. We use the theory of semigroups in conjunction with spectral decompositions of their generators in order to derive detailed representation formulas for shooting operators in function space and their adjoints. A spectral truncation approach delivers a self-adjoint indefinite Newton-Picard preconditioner for the saddle-point system of optimality conditions in function space. We show that this preconditioner leads to convergence in a function space fixed-point iteration. Moreover, we discuss that this preconditioner can be approximated well by a two-grid approach. We address some implementation issues and present numerical results for three-dimensional instationary problems with more than 100,000,000 degrees of freedom.