Low-rank and sparse dynamic mode decomposition

Even though fluid flows possess an exceedingly high number of degrees of freedom, their dynamics often can be approximated reliably by models of low complexity. This observation has given rise to the notion of coherent structures – organized fluid elements that, together with dynamic processes, are responsible for the bulk of momentum and energy transfer in the flow. Recent decades have seen great advances in the extraction of coherent structures from experiments and numerical simulations. Proper orthogonal de-composition (POD) modes (Lumley 2007; Sirovich 1987), global eigenmodes, frequential modes (Sipp et al. 2010), and balanced modes (Moore 1981; Rowley 2005) have provided useful insight into the dynamics of fluid flows. Recently, dynamic mode decomposition (DMD) (Rowley et al. 2009; Schmid 2010) has joined the group of feature extraction techniques. Both POD and DMD are snapshot-based post-processing algorithms which may be applied equally well to data from simulations or experiments. By enforcing or-thogonality, POD modes possess multi-frequential temporal content; on the other hand, DMD modes are characterized by a single temporal frequency. DMD modes may poten-tially be non-normal, but this non-normality may be essential to capturing certain type