Data driven modal decompositions: analysis and enhancements
For researchers using DMD in fluid dynamics and nonlinear dynamics, this work offers incremental enhancements to improve reliability and accuracy of spectral analysis.
The paper improves Dynamic Mode Decomposition (DMD) by introducing a data-driven residual formula for selecting Ritz pairs and adapting Ritz vector refinement, leading to more precise spectral information. Numerical experiments demonstrate advantages of the proposed DDMD_RRR method.
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).