A semi-continuous formulation for goal-oriented reduced-order models

Modern computational and experimental techniques can represent the detailed dynamics of complex systems using large numbers of degrees of freedom. To facilitate human interpretation or the optimal design of control systems, however, reduced-order models (ROMs) are required. Conventional reduced-order modeling techniques, such as those based on proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (BPOD), and dynamic mode decomposition (DMD), are purely data-driven. That is, the governing equations are not taken into account when determining the solution basis of the ROM. The resulting ROMs are thus sub-optimal, particularly when low numbers of degrees of freedom are used. Bui-Thanh et al. addressed this problem by determining ROM solution bases using a goal-oriented optimization procedure that seeks to minimize the error between the full and reduced-order goal functionals with the reduced-order model as a constraint. However, several issues limit the application of this approach. First, it requires explicit input matrices with the dimension of the reference data that result from spatially discretizing the governing equations. In addition, its derivation is restricted to linear governing equations and goal functionals. To overcome these limitations, our research group has proposed an alternative, a semi-continuous formulation (SCF), in which the ROM constraint and the optimization process are defined in a continuous setting. In this thesis, the mathematical framework of the SCF is illustrated, as is the algorithm used to solve the optimization problem.