Multiresolution Dynamic Mode Decomposition

Abstract

We demonstrate that the integration of the recently developed dynamic mode decomposition (DMD) with a multi-resolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy of multi-resolution 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 multi-resolution dynamic mode decomposition 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 multi-resolution dynamic mode decomposition (mrDMD) method is demonstrated on several examples involving multi-scale dynamical data, showing excellent decomposition results, including sifting the El Niño 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.

Methods for data-driven multiscale model discovery for materials

“…A remarkable feature of the DMD algorithm is its modularity for mathematical enhancements. Specifically, the DMD algorithm can be engineered to exploit sparse sampling [61,62], it can be modified to handle inputs and actuation [71], it can be used to more accurately approximate the Koopman operator when using judiciously chosen functions of the state-space [72], it can be easily made computationally scalable [73], and it can easily decompose data into multiscale temporal features in order to produce a multi-resolution DMD (mrDMD) decomposition [74]. Few mathematical architectures are capable of seamlessly integrating such diverse modifications of the dynamical system.…”

Movement correction in DCE-MRI through windowed and reconstruction dynamic mode decomposition

“…Even though the low-rank and sparse representations of an image sequence have been reported for DMD [7,12], the method that we propose here is essentially different. In [7], the authors exploit the low-rank and sparse representation within each frame.…”

“…Specifically, low rank revealing the background and sparse presenting the foreground of that particular frame. Recently, in [12] DMD with a multi-resolution approach (MR-DMD) decomposed video streams into multi-time scale features and objects. The MR-DMD approach is similar to that of applying standard DMD technique at several resolutions after discarding the slow varying modes (background modes or the most significant modes).…”

“…In order to select the most significant dynamic modes, the method suggested in [7,12] is to calculate the logarithmic values of the diag(σ ). The frequencies which are near origin are the most significant modes.…”

“…Specifically, the parts in image sequence that do not change in time have an associated frequency μ j ≈ 0, which corresponds to background. DMD with a multi-resolution [12] approach decomposes an image sequence into multi-time scale features and objects. The MR-DMD approach is similar to that of applying standard DMD technique at several resolutions on fast varying dynamic modes after discarding the background modes or the slow varying modes.…”