In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the is expensive, however, and therefore often impossible in model-predicti...

PDE-constrained optimization problems involving inequality constraints for the design variables are considered. The optimization problems and hence their solutions are subject to perturbations in the data. An output functional (quantity of interest) is given which depends on both the optimal state and design variables. Conditions are derived such that the quantity of interest at the optimal sol...

The main part of this chapter is devoted to the development and presentation of a coupled multigrid method for the solution of saddle point systems (2.51) arising from the discretization of PDE constrained optimization problems. In subsequent chapters, the devised method will be adapted to handle inequality constraints on the control, and it will be employed for the solution of the systems, whi...

This paper describes a framework for the design of microvascular polymeric components for active cooling applications. The design of the embedded networks involves complex and competing objectives that are associated with various physical processes. The optimization tool includes a PDE solver based on advanced finite element techniques coupled to a multi-objective constrained genetic algorithm....

We propose a new non-conforming localized model reduction paradigm for efficient solution of large scale or multiscale PDE constrained optimization problems. The new conceptual approach goes beyond the classical offline/online splitting of traditional projection based model order reduction approaches for the underlying state equation, such as the reduced basis method. Instead of first construct...

A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control p...

Shape optimization based on shape calculus has received a lot of attention in recent years, particularly regarding the development, analysis, and modification efficient algorithms. In this paper we propose investigate nonlinear conjugate gradient methods Steklov-Poincar\'e-type metrics for solution problems constrained by partial differential equations. We embed these into general algorithmic f...