A Parametric and Feasibility Study for Data Sampling of the Dynamic Mode Decomposition--Part II: Input Variable, Truncation, Interpolation, and Cross-Validation.

This serial effort investigates the nuances in the sampling of the Dynamic Mode Decomposition (DMD) with an engineering-oriented emphasis. In Part I, the parametric study established the convergence of sampling range and resolution. In this Part II, we examined the effects of the input variable, truncation, and data interpolation on the integrity and output of the DMD. Results showed that the selection of input variable, even when the fluctuating, mean, or instantaneous velocity fields root from the same fluid mechanical origin, significantly affects the dynamical content of the input series, thus the aptness of the DMD modeling. We proved that the DMD yields the optimal Koopman representation when the input system appeals to clear periodicity, such as turbulence. Truncation, though frequently applied for the Proper Orthogonal Decomposition (POD), can radically shift the stability and causality of the linearly time-invariant (LTI) representation the DMD generates. Results also reinforced the existence of dynamical low-energy POD states. A small degree of truncation maintains the integrity of low-order dynamics but sacrifices reconstruction accuracy, and it is the inverse when the truncation order is substantial. Pre-DMD data interpolation inevitably introduces numerical noise. However, the DMD can distinguish the pseudo-dynamics from a system's intrinsic dynamics with the tested high-order interpolation scheme. Finally, we cross-validated previous findings with the cylinder wake, proving the universality and generality of the convergence states, namely the Initialization, Transition, Stabilization, and Divergence. Bi-parametric cross-validation reinforced the mutual independence in the convergence of range and resolution and supported the practical recommendation of 15 frames per cycle.