Reduced-order methods have been used to reliably solve discrete parametrized mathematical models and to lessen the associated computational complexity of these models by projecting them onto spaces of reduced order. The reduced-order spaces are spanned by a finite number of full-order solutions, a reduced basis, that, if well-chosen, provide a good approximation of the entire solution manifold. Reduced-order methods have been used with various problem classes including different types of constrained parametrized problems such as constrained parametrized partial differential equations (PDEs) where the constraints are built into the PDEs, and parametrized PDE-constrained optimization problems, or PDE-control problems, where the constraints themselves are PDEs. In the deterministic setting, both of these problem classes involve discrete models that are of saddle-point form and can be computationally expensive to solve. It is well known that saddle-point problems must satisfy an inf-sup condition to ensure stability of the solution, thus, solving deterministic variations of these models requires the consideration and satisfaction of an inf-sup stability condition. When these models are subject to parametrization, the solution to a deterministic problem is sought for many parameter values. Reduced-order models for these problem classes are often constructed so that they mirror the full-order models and are also of saddle-point form. In the established RB methods we study for the problem classes explored in this thesis, the reduced basis is represented as a block-diagonal matrix that produces a saddle-point reduced system and is augmented to satisfy inf-sup stability. Two methods of building an augmented RB to ensure inf-sup stability that have been well studied are augmentation by aggregation and augmentation by the supremizer. We present a comparative study of these two common methods of stabilizing reduced order models, through use of the supremizer and through aggregation, and compare the accuracy, efficiency, and computational costs associated with them for solving the parametrized PDE-control problem.

We propose a new approach to implementing the RB basis, the stacked reduced basis, that produces a reduced system that is not of saddle-point form. Implementing the stacked reduced basis avoids the necessity to satisfy the inf-sup condition to ensure stability and therefore, to augment the reduced bases spaces. This results in a reduced basis system of smaller order, which reduces the computational work in the online step. While inf-sup stability is avoided, there are still issues with the stability of the stacked reduced system during RB construction, particularly for the constrained PDE problem class. We show that this can be addressed by penalization and present results to show that penalization improves the stability of an established augmented RB method as well. We present numerical results to compare the new approach to two developed ways of implementing the RB method (with both commonly accepted choices of augmentation) and prove the efficiency of the proposed approach. We study the efficiency of the stacked reduced basis for both PDE-control problems and constrained PDEs subject to parametrization.