Multiresolution Dynamic Mode Decomposition

We demonstrate that the integration of the recently developed dynamic mode decomposition (DMD) with a multiresolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy of multiresolution time-scale components. A one-level separation allows for background (low-rank) and foreground (sparse) separation of dynamical data, or robust principal component analysis. The multiresolution DMD (mrDMD) is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. DMD modes with temporal frequencies near the origin (zero-modes) are interpreted as background (low-rank) portions of the given dynamics, and the terms with temporal frequencies bounded away from the origin are their sparse counterparts. The mrDMD method is demonstrated on several examples involving multiscale dynamical data, showing excellent decomposition results, including sifting the El Nin͂o mode from ocean temperature data. It is further applied to decompose a video data set into separate objects moving at different rates against a slowly varying background. These examples show that the decomposition is an effective dynamical systems tool for data-driven discovery.