Optimized Sampling for Multiscale Dynamics

TL;DR

This paper introduces a multiresolution Dynamic Mode Decomposition method that enables efficient sparse sampling and accurate analysis of complex multiscale dynamical systems without requiring full data, demonstrated on climate data.

Contribution

The paper presents a novel multiresolution DMD approach with optimized sensor placement using QR pivots for efficient multiscale dynamics analysis.

Findings

Accurate reconstruction of sea-surface temperature with few sensors.

Effective classification of El Niño events using sparse data.

Multiresolution DMD captures nonlinear dynamics without explicit equations.

Abstract

The characterization of intermittent, multiscale and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determined sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Ni\~no events.