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Automated extension of fixed point pde solvers for optimal design with bounded retardation

Abstract:

We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.