A Survey on the Methods and Results of Data-Driven Koopman Analysis in the Visualization of Dynamical Systems

Abstract

Koopman mode decomposition is a flow analysis technique developed by Igor Mezic in 2004, based upon the Koopman operator first proposed by Bernard Koopman in 1931. Via Koopman decomposition any non-chaotic well-sampled dynamic system - linear, non-linear, laminar or turbulent - is broken down into single-frequency repetitive components (modes). This paper presents a survey consolidating published information regarding data-driven Koopman analysis techniques. It is intended to aid researchers exploring the suitability of data-driven Koopman analysis in anticipation of developing their own modeling. A basic mathematical explanation of Koopman analysis is given with emphasis toward the data-driven Dynamic Mode Decomposition (DMD) solution, which converges to the Koopman operator given a highly-sampled dataset. The four primary uses of Koopman analysis: flow analysis, power grid analysis, building thermal analysis, and biomedical analysis are discussed, along with other publications. Finally, weaknesses and problems inherent within Koopman analysis/DMD will be enumerated, alongside potential solutions. Koopman analysis is a computationally complex, yet often suitable method for determining periodic motion in any highly-sampled dataset. When compared to a similar analysis method, Proper Orthogonal Decomposition, Koopman analysis often provides additional detail regarding the structure of less significant modes present, albeit at the cost of increased computational complexity.