Ph.D Thesis


Ph.D StudentCohen Ido
SubjectSpectral Analysis of Nonlinear Flows
DepartmentDepartment of Electrical and Computer Engineering
Supervisor ASSOCIATE PROF. Guy Gilboa


Abstract

This work provides a new approach to nonlinear flow analysis, based on linear and nonlinear spectral theory. The findings can be applied in various domains, such as optimization, image processing, neural networks, control systems, and computational physics.


First, we present a flow to solve nonlinear eigenpair problems commonly found in nonlinear physical models. The method generalizes the min/max Rayleigh quotient process, often used for these problems. The suggested flow converges at eigenpairs and is very robust to noise and to perturbations in the initial conditions, compared to conventional methods.


We then present a signal processing framework, based on Partial Differential Equations (PDEs), derived by g-homogeneous operators, where 0≤g<1. This framework includes transform (decomposition), inverse-transform (reconstruction), filtering, and spectrum adhering to Parseval-type equality. The proposed approach generalizes the spectral total-variation framework, and uses, for the first time in this context, fractional calculus theory. The method is applied for image decomposition, based on the p-Laplacian spectra.


A discrete version of the above framework is suggested by applying a widely used fluid dynamics analysis tool, Dynamic Mode Decomposition (DMD). Applying this tool naively reveals an inherent problem in DMD, we term as the DMD paradox. We give a lower bound of the DMD reconstruction error, in a certain fundamental case. We show that this type of error appears in any flow that does not decay exponentially. We suggest a new algorithm that solves the DMD paradox and illustrates its validity on synthetic and natural data. 


Finally, we extend our analysis of DMD using Koopman operator theory. Conditions for the existence of Koopman eigenfunctions are provided. The dynamics are represented as a curve in high dimensional space. We show relations between the curve geometry and the definitions of observability and controllability in control theory. We propose a new mode decomposition approach, based on the decay profile typical to the dynamics. Insights and possible uses of the above findings are discussed.