Residual Dynamic Mode Decomposition: Robust and verified Koopmanism

Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a major challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes, and dealing with continuous spectra, both of which occur regularly in turbulent flows. Residual Dynamic Mode Decomposition (ResDMD), introduced by (Colbrook & Townsend 2021), overcomes some of these challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in a variety of fluid dynamic situations, at varying Reynolds numbers, arising from both numerical and experimental data. Examples include: vortex shedding behind a cylinder; hot-wire data acquired in a turbulent boundary layer; particle image velocimetry data focusing on a wall-jet flow; and acoustic pressure signals of laser-induced plasma. We present some advantages of ResDMD, namely, the ability to verifiably resolve non-linear, transient modes, and spectral calculation with reduced broadening effects. We also discuss how a new modal ordering based on residuals enables greater accuracy with a smaller dictionary than the traditional modulus ordering. This paves the way for greater dynamic compression of large datasets without sacrificing accuracy.