Model order reduction in the frequency domain via spectral proper orthogonal decomposition

In this thesis, we introduce a novel model order reduction framework for harmonically and randomly forced dynamical systems. Specifically, we emphasize the usage of spectral proper orthogonal decomposition (SPOD), recently revived by Towne et. al. (2018), which results in sets of orthogonal modes, each oscillating at a single frequency, that are said to optimally represent coherent-structures evolving in space and time. However, reduced-order models (ROMs) using SPOD modes have not yet been developed. Hence, in this study we investigate the potential of a novel approach utilizing SPOD modes to construct the lower-dimensional subspace for ROMs. Upon the discrete-time Fourier-transform (DFT) of the governing ordinary differential equation (ODE) system, an orthogonal projection onto the SPOD modes is performed, analogous to Proper Orthogonal Decomposition (POD) Galerkin ROMs, but compressing the system at each discrete frequency within the frequency domain. The ROM is solved at each frequency and after the inverse DFT of the spectral solution matrix we obtain the entire solution for a given timespan at once, with no time integration necessary. This new approach is illustrated using the example PDE of steady passive scalar transport in an inhomogenous, time-invariant flow field. Finally, we compare the performance of our ROM with the standard POD-Galerkin ROM in terms of accuracy and computational speedup