Abstract
This paper describes a general approach for the construction of approximations of the Koopman operator associated with deterministic continuous semiflows over a complete metric space X. A primary contri- bution of the paper is the derivation of rates of convergence for approximations of Koopman operators for large classes of evolution equations. The rates of convergence in the paper are derived in two substantially different scenarios. If the state space X is compact, we consider when the samples Ξ are dense in the entire state space X. We also study cases when samples are dense in a limiting subset Ω ⊂ X that is a proper subset of X. Two general classes of methods are described in the paper, referred to as intrinsic and extrinsic methods. Intrinsic methods define bases of approximation from kernels that are defined in terms of, or having knowledge of, the limiting set Ω. Extrinsic methods use kernels that do not depend on the knowledge of the limiting set Ω. In both types of approximations, the regularity of the underlying set and the smoothness of the space of functions on which the Koopman operator acts determine the rate of approximation. In the strongest error bounds derived in the paper, it is shown that the error in approximation of the Koopman operator decays like O(hΩn,Ω)p where hΩn,Ω is the fill rate of the samples Ωn in the limiting set Ω and p is an exponent related to the choice of the kernel and the smoothness of functions on which the Koopman operator acts. Such error bounds are obtained when either the limiting subset Ω = X, when it is a proper subset Ω ⊂ X that is sufficiently regular, or when it is a type of smooth manifold Ω = M.
Sai Tej Paruchuri
Postdoctoral Research Associate in Plasma Control
My research interests include plasma control, dynamics and controls, vibrations and adaptive structures, data-driven modeling.