Large dynamical systems with complex behavior, such as those arising in climate and weather modeling have over the years been treated with a variety of approaches from both deterministic and probabilistic perspective, perhaps the most expansive of these being Lorenz' 1963 work that lead to a massive amounth of work both within the field and outside of it. In this talk we advocate an approach that stems from von Neumann's and Koopman's work on operator methods in classical mechanics, where the key object is not the phase-space representation of the system and the associated geometry (e.g. of the attractors), but an operator representing the system on the space of observables, called the Koopman operator.
Two topics are treated within this context:
The first is the issue of decomposition of dynamics into "natural modes". The so-called Proper Orthogonal Decomposition (POD) (or Karhunen-Loeve, Singular Value Decomposition) is a popular method of achieving this. We study the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system within the context of the spectral properties of aforementioned Koopman operator. We apply this theory to obtain a decomposition of the process. This allows us to extract the {\it almost periodic} part of the evolving process. The remainder of the process has continuous spectrum and can be decomposed using POD. We call this the Mixed Orthogonal Decomposition (MOD).
The second is the issue of uncertainty of process outputs. We define a notion of uncertainty specifically designed for the fact that the process is described as a dynamical system. Specifically, a dynamical system that asymptotes to a stable equilibrium point from any initial condition has asymptotically a perfectly certain output. The notion of uncertainty that we propose measures deviations from that case. We discuss differences of this measure and other measures of uncertainty such as variance and entropy, and its connection with the properties of the Koopman operator.
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