Taming the Butterfly Effect -- Towards Computational Engineering of Unsteady and Chaotic Systems

Many dynamical systems of scientific and engineering importance are unsteady and chaotic. These systems can be found in unsteady fluid flows around airplanes and inside gas turbine engines, aero-elastic oscillations, our climate system and molecular dynamics. This talk presents an adventure beyond just performing simulation of chaotic systems, towards aspects of computational engineering. These include optimization, control, uncertainty quantification, and data based inference. Our first focus is on sensitivity analysis, a widely used tool in computational engineering that calculates derivatives of simulation outputs to input parameters.

We show that existing methods for sensitivity analysis often fail in chaotic systems. The inability to exchange a limit and a derivative is the mathematical reason of this failure. A new computational algorithms for solving this challenge is the "least squares shadowing" method. We show many interesting aspects of this new method. We present application of this new method to several chaotic dynamical systems, including preliminary results on chaotic fluid flow systems and aeroelastic oscillations.