Temporal-Harmonic Specific POD Mode Extraction

Abstract

While POD / PCA / KL approximations are statistically energetically optimal, statistical optimality is indeed the sole consideration these (equivalent) methods invoke. This type of approximation is neither geared for, nor is it optimized to extract modes based on their significance to an underlying system dynamics. Furthermore, as computational considerations limit the size of empirical ensembles used for mode extraction, the resulting mode set is significantly effected by the arbitrariness of the ensemble selection. System theoretic model reductions methods aim to home on dynamically significant modes by direct interrogation of the underlying equation, such as the linearized Navier-Stokes equations. An alternative / complimentary approach is to impose a priori knowledge of structural properties, such as symmetry and periodicity, on the mode-extraction procedure. The idea is that these conditions will force the selection of physically meaningful modes, and thus enables an effective appeal to first principles. Here we focus on systems known to be periodically dominant, and describe a simple method to extract modes associated with temporal harmonics. The method accommodates time variations in the dominant frequency(ies) and exploits a preliminary data compression, such as by the standard POD procedure.