Nonlinear Optimal Control and Design Optimization: a comparison of Direct-adjoint-Looping and one-shot methods

In our previous work we have considered  PDE-constrained optimization for steady-state problems using i) Direct-Adjoint-Looping (DAL), and ii) one-shot methods. We now extend such analysis to transient problems. For the DAL method, the forward problem, described by nonlinear PDES  (e.g., unsteady Reynolds-Averaged-Navier-Stokes (URANS) or Kuramoto–Sivashinsky (KS) equation, which can be chaotic) is first solved for the given design/control variables. We then derive the continuous adjoint equations, which are solved backward in time. We discuss the checkpointing method, and some variations of it, to handle the backward solution of adjoint equations  in time. We also discuss the impact of the step-size on the convergence rate. In the one-shot method, we couple the adjoint solver with the forward solver and updates of the control variable and we use an approximate Hessian to speed up the convergence rate. We demonstrate the limitations of each method for various classes of  problems and cost functionals.