Multi-scale Uncertainty Propagation in Dynamical Systems

Theoretical and computational methods to analyze and control the dynamic behavior of complex systems under uncertainty were investigated. Compressive Polynomial Chaos Expansions were used to circumvent the large-scale difficulties common in other Polynomial Chaos expansions. In the area of Koopman and Dynamic Mode Decomposition Analysis, stable and efficient computational techniques were developed that address a suite of problems, from Ergodic Quotient computations to complex turbulent flow characterizations. This resulted in a Koopman mode theory that rigorously unifies a number of seemingly distinct concepts advanced in fluid dynamics. Using the setting of stochastic structured uncertainty, a purely input-output theory of systems with time-varying stochastic parameters was developed. New mean-square stability tests were discovered with two important features, computational complexity that scales with number of uncertainties rather than with state dimension, and the ability to handle correlated uncertainty. Distributed control design in large-scale stochastic networks was studied. In the limit of large system size, surprising dimensionality dependencies and phase transition phenomena were discovered in the optimal control design problem itself.