Optimization problems constrained by partial differential equations are ubiquitous in modern science and engineering. They play a central role in optimal design and control of multiphysics systems, as well as nondestructive evaluation and detection, and inverse problems. Methods to solve these optimization problems rely on, potentially many, numerical solutions of the underlying equations. For complicated physical interactions taking place on complex domains, these solutions will be computationally-expensive – in terms of both time and resources – to obtain, rendering the optimization procedure difficult or intractable.
 
I will introduce a globally convergent, non-quadratic trust-region method to accelerate the solution of PDE-constrained optimization problems by adaptively reducing the dimensionality of the underlying computational physics discretization. In this approach, the method of snapshots and Proper Orthogonal Decomposition (POD) are used to build a reduced-order model whose fidelity is progressively enriched while converging to the optimal solution. This ensures the reduced-order model is trained exactly along the optimization trajectory and effort is not wasted by training in other regions of the parameter space. A novel minimum-residual framework for computing surrogate sensitivities of the reduced-order model is introduced that equips the trust-region method with desirable properties. The proposed method is shown to solve canonical aerodynamic shape optimization problems several times faster than accepted methods. This work has been extended to address the specific challenges posed by topology optimization, where high-dimensional parameter spaces are inevitable.