Optimal Design with Bounded Retardation for Problems with Non-separable Adjoints
For many real world processes there are mathematical models in terms of PDEs. However, for most of the applications any analytical solution is out of reach. Therefore, numerous simulation codes depending on some design parameters have been developed and implemented. In our research we focus on the transition from such simulation codes to optimization, where the design parameters are chosen in such a way that the underlying model is optimal with respect to some measure. In contrast to general non-linear programming we assume that the models are too large for the direct evaluation and factorization of the occurring derivative matrices but that a slowly convergent fixed-point iteration is available to compute a solution of the model for fixed parameters. Therefore, we pursue the so-called One-shot approach, where the forward simulation is complemented with an adjoint iteration, which can be obtained by handcoding, the use of Automatic Differentiation techniques, or a combination thereof. The resulting adjoint solver is then coupled with the primal fixed-point iteration and an optimization step for the design parameters to obtain an optimal solution of the problem. To guarantee the convergence of the method an appropriate sequencing of these three steps, which can be applied either in a parallel (Jacobi) or in a sequential (Seidel) way,and a suitable choice of the preconditioner for the design step are necessary. In this paper we give a brief overview of our research results for both approaches and a choice for the \textit{right} design preconditioner, i.e. we present a preconditioner motivated by the Hessian of a doubly augmented Lagrangian and the projected Hessian of the Lagrangian for the Jacobi- and the Seidel-Version, respectively. Furthermore, we consider the extension of the One-shot approach to the infinite dimensional case and problems with unsteady PDE constraints. The theoretical results are applied to several real-world applications. In particular, we present two models from marine science and aerodynamic shape optimization.