New methods for sensitivity analysis of chaotic dynamical systems

Abstract

Computational methods for sensitivity analysis are invaluable tools for fluid dynamics research and engineering design. These methods are used in many applications, including aerodynamic shape optimization and adaptive grid refinement. However, traditional sensitivity analysis methods break down when applied to long-time averaged quantities in chaotic dynamical systems, such as those obtained from high-fidelity turbulence simulations. Also, a number of dynamical properties of chaotic systems, most notably the "Butterfly Effect", make the formulation of new sensitivity analysis methods difficult. This paper will discuss two chaotic sensitivity analysis methods and demonstrate them on several chaotic dynamical systems including the Lorenz equations and a chaotic Partial Differential Equation, the Kuramoto-Sivshinsky equation. The first method, the probability density adjoint method, forms a probability density function on the strange attractor associated with the system and uses its adjoint to find gradients. This was achieved using a novel numerical method in which the attractor manifold, instead of a region of phase space, is discretized. The second method, the Least Squares Sensitivity method, finds some "shadow trajectory" in phase space for which perturbations do not grow exponentially. This method is formulated as a quadratic programing problem with linear constraints. Several multigrid-in-time methods to solve the KKT system arising from this optimization problem will be discussed in depth. While the probability density adjoint method is better suited for smaller systems and reduced order models, least squares sensitivity analysis, solved with a multigrid-in-time method could be applied to higher dimensional systems such as high fidelity fluid flow simulations.