UC San Diego

Ranging from natural phenomena such as biological and chemical systems to artificial technologies such as mechanical and electronic devices, dynamical systems form an inseparable part of the world surrounding us. Understanding, modeling, predicting, and controlling such systems have always been the leading goals of science and engineering. While in the past centuries, the most advances in the field of dynamical systems were mainly analytical and based on limited observations, in the last decade, we have witnessed a rapid growth in our ability to gather, store, and process data. This data-driven revolution has imposed a high demand for new viewpoints and systematic structures that can effectively utilize the available modern tools. The Koopman operator theory for dynamical systems has recently gained widespread attention. Unlike the traditional state space methods, which explain the evolution of the states according to the dynamics, the Koopman operator characterizes the effect of the dynamics on functions in a linear function space. Despite the system being linear or nonlinear, its associated Koopman operator is always linear. This linearity proves to be extremely useful for algorithmic computations.

This dissertation is focused on the data-driven analysis of dynamical systems based on their associated Koopman operator. Given the infinite-dimensional nature of the Koopman operator, we aim to find finite-dimensional spaces on which the operator's action can be captured accurately. Such finite-dimensional spaces must be close to being invariant under the action of the Koopman operator; otherwise, the approximation will be erroneous. We provide data-driven algebraic methods to provably find the exact maximal Koopman-invariant subspace and all Koopman eigenfunctions in any arbitrary finite-dimensional space of functions. We also offer equivalent algorithms tailored for working with large and streaming data sets, as well as parallel computing hardware. Keeping in mind that an exact finite-dimensional linear model might not capture complete information for some systems, we also consider approximating subspaces to capture more information about the dynamics. We provide data-driven measures to quantify how close a linear space of functions is to being invariant under the Koopman operator. In addition, we provide an algebraic procedure that can approximate Koopman-invariant subspaces with tunable accuracy.