Residual Dynamic Mode Decomposition: Robust and verified Koopmanism

Abstract

Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a major challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes, and dealing with continuous spectra, both of which occur regularly in turbulent flows. Residual Dynamic Mode Decomposition (ResDMD), introduced by (Colbrook & Townsend 2021), overcomes some of these challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in a variety of fluid dynamic situations, at varying Reynolds numbers, arising from both numerical and experimental data. Examples include: vortex shedding behind a cylinder; hot-wire data acquired in a turbulent boundary layer; particle image velocimetry data focusing on a wall-jet flow; and acoustic pressure signals of laser-induced plasma. We present some advantages of ResDMD, namely, the ability to verifiably resolve non-linear, transient modes, and spectral calculation with reduced broadening effects. We also discuss how a new modal ordering based on residuals enables greater accuracy with a smaller dictionary than the traditional modulus ordering. This paves the way for greater dynamic compression of large datasets without sacrificing accuracy.

References in corpus (22)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• On Dynamic Mode Decomposition: Theory and Applications
• Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
• Sparsity-promoting dynamic mode decomposition
• Applied Koopmanism
• Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
• Nonlinear mode decomposition with convolutional neural networks for fluid dynamics
• Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
• Study of dynamics in post-transient flows using Koopman mode decomposition
• Data-driven resolvent analysis
• Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
• On the Structure of Time-delay Embedding in Linear Models of Non-linear Dynamical Systems
• Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)
• Koopman analysis of the long-term evolution in a turbulent convection cell
• Reproducing kernel Hilbert space compactification of unitary evolution groups
• Koopman mode expansions between simple invariant solutions
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Koopman analysis of Burgers equation
• On the infinite-dimensional QR algorithm

Cited by in corpus (10)

• Vehicular Applications of Koopman Operator Theory -- A Survey
• Koopman operator learning using invertible neural networks
• An overview of Koopman-based control: From error bounds to closed-loop guarantees
• Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds
• Real-Time Motion Detection Using Dynamic Mode Decomposition
• Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces
• Group-Convolutional Extended Dynamic Mode Decomposition
• Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
• Dynamic mode decomposition for detecting oscillatory transient activity via sparsity and smoothness regularization
• Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine