Multi-Resolution Dynamic Mode Decomposition

TL;DR

Multi-Resolution Dynamic Mode Decomposition (mrDMD) is introduced as a robust, multi-scale, data-driven method for decomposing complex nonlinear dynamical systems into hierarchical time-scale components without requiring explicit equations.

Contribution

The paper presents a novel multi-resolution extension of DMD that effectively separates multi-scale dynamics and background/foreground components in complex systems.

Findings

Successfully decomposes ocean temperature data to isolate El Niño modes

Effectively separates moving objects from backgrounds in video data

Demonstrates robustness and accuracy in multi-scale dynamical data analysis

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 Ni\~no 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.