  To solve optimization problems with parabolic PDE constraints, often methods
working on the reduced objective functional are used. They are computationally
expensive due to the necessity of solving both the state equation and a
backward-in-time adjoint equation to evaluate the reduced gradient in each
iteration of the optimization method. In this study, we investigate the use of
the parallel-in-time method PFASST in the setting of PDE constrained
optimization. In order to develop an efficient fully time-parallel algorithm we
discuss different options for applying PFASST to adjoint gradient computation,
including the possibility of doing PFASST iterations on both the state and
adjoint equations simultaneously. We also explore the additional gains in
efficiency from reusing information from previous optimization iterations when
solving each equation. Numerical results for both a linear and a non-linear
reaction-diffusion optimal control problem demonstrate the parallel speedup and
efficiency of different approaches.
