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 decomposition (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 potentially be non-normal, but this non-normality may be essential to capturing certain types of dynamical effects. For an in-depth discussion of the connection between DMD and other data decomposition methods, please refer to Schmid (2010). By projecting the full system onto the extracted modes, the governing equations may be replaced by a dynamical system with fewer degrees of freedom. This facilitates computa-tionally tractable investigations of stability or receptivity as well as a model-based control design. In many situations, however, it is not trivial to identify a subset of modes that have the strongest impact on the flow dynamics. For example, spatial non-orthogonality of the DMD modes may introduce a poor quality of approximation of experimentally or numerically generated snapshots when only a subset of modes with the largest amplitude is retained. Recent attempts at extracting only a subset of desired frequencies and basis vectors (Chen et al. 2012) rely on a non-convex optimization problem whose solution in general requires an intractable combinatorial search. In order to strike a balance between the quality of approximation (in the least-squares sense) and the number of modes that are used to approximate the given fields, this brief develops low-rank and sparsity-promoting versions of the standard DMD algorithm. To achieve this objective, we combine tools and ideas from linear algebra and convex optimization with the emerging area of compressive sensing (Candès & …