An Investigation into Koopman Operator Theory: Linear Subspace Methods for Koopman Mode Decomposition, and Parameter Estimation for Nonlinear Discrete Time Dynamical Systems using Koopman Operator Theory
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This thesis is about Koopman operator theory. The first, main, chapter of this thesis concerns the connection between classical linear subspace methods and Koopman operator theory. We show that these are very much related. The Univariate Principal Components (UPC) algorithm can be used to construct approximations of Koopman eigenfunctions, lying within the span of delay coordinates. In general these are poor eigenfunction approximation, but provide a good Koopman mode decomposition, when compared to Extended Dynamic Mode Decomposition (EDMD) on several low-dimensional examples. The short second chapter concerns an investigation into the usage and limitations of parameter estimation of discrete-time dynamical systems from partial measurements using Koopman operator theory. We argue that including delay coordinates allows for the estimation of all parameters, but that doing so gives rise to a computational issue in computing Galerkin inner products. We propose to overcome this by continuous approximation of the empirical distribution. We are not able to substantiate these ideas with numerical results. We find that the computational costs are too high, and propose to tackle this by resorting to polynomial approximation, for which integration and optimization are easier.