Optimal linear model reduction using SPOD modes

Spectral proper orthogonal decomposition (SPOD) modes provide an optimal linear representation of the long-term evolution of stationary flows as measured by a space-time norm. In other words, the truncated SPOD representation of a trajectory becomes more accurate, on average, than the representation in any other space-time basis as the time interval becomes long. We present a method to solve for the exact SPOD coefficients that represent a trajectory in forced linear systems, thereby obtaining this optimal representation given the initial condition and forcing. The method works by projecting the unreduced equations in the frequency domain onto the SPOD modes at each frequency, and may be formulated as a frequency domain Petrov-Galerkin method. The method requires an ensemble of realizations of the system from which to calculate the SPOD modes and the entire time series of the forcing for the realization to be calculated. With these inputs, the method is observed to be substantially more accurate than standard methods, such as POD-Galerkin and balanced truncation, which is expected given the SPOD optimality property. It also scales favorably to large systems, and is significantly faster than standard linear model reduction methods.