  Parallel-in-time methods are developed to accelerate the direct-adjoint
looping procedure. Particularly, we utilize the Paraexp algorithm, previously
developed to integrate equations forward in time, to accelerate the
direct-adjoint looping that arises from gradient-based optimization. We
consider both linear and non-linear governing equations and exploit the linear,
time-varying nature of the adjoint equations. Gains in efficiency are seen
across all cases, showing that a Paraexp based parallel-in-time approach is
feasible for the acceleration of direct-adjoint studies. This signifies a
possible approach to further increase the run-time performance for optimization
studies that either cannot be parallelized in space or are at their limit of
efficiency gains for a parallel-in-space approach.
