Data driven modal decompositions: analysis and enhancements

TL;DR

This paper introduces an improved data-driven Dynamic Mode Decomposition method, DDMD_RRR, enhancing spectral accuracy and reliability for analyzing nonlinear dynamical systems and fluid flows.

Contribution

It develops a new residual-based selection of Ritz pairs and refines Ritz vectors within a generalized weighted inner product framework for better spectral analysis.

Findings

Enhanced spectral accuracy with residual-based Ritz pair selection

Improved reliability and robustness of DMD in data-driven scenarios

Numerical experiments demonstrate advantages of the proposed DDMD_RRR method

Abstract

The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of fluid flows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers further improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments are used to illustrate the advantages of the proposed method, designated as DDMD_RRR (Refined Rayleigh Ritz Data Driven Modal Decomposition).