Efficient sensitivity analysis of chaotic systems and applications to control and data assimilation

Abstract

Sensitivity analysis is indispensable for aeronautical engineering applications that require optimisation, such as flow control and aircraft design. The adjoint method is the standard approach for sensitivity analysis, but it cannot be used for chaotic systems. This is due to the high sensitivity of the system trajectory to input perturbations; a characteristic of many turbulent systems. Although the instantaneous outputs are sensitive to input perturbations, the sensitivities of time-averaged outputs are well-defined for uniformly hyperbolic systems, but existing methods to compute them cannot be used. Recently, a set of alternative approaches based on the shadowing property of dynamical systems was proposed to compute sensitivities. These approaches are computationally expensive, however. In this thesis, the Multiple Shooting Shadowing (MSS) [1] approach is used, and the main aim is to develop computational tools to allow for the implementation of MSS to large systems. The major contributor to the cost of MSS is the solution of a linear matrix system. The matrix has a large condition number, and this leads to very slow convergence rates for existing iterative solvers. A preconditioner was derived to suppress the condition number, thereby accelerating the convergence rate. It was demonstrated that for the chaotic 1D Kuramoto Sivashinsky equation (KSE), the rate of convergence was almost independent of the #DOF and the trajectory length. Most importantly, the developed solution method relies only on matrix-vector products. The adjoint version of the preconditioned MSS algorithm was then coupled with a gradient descent method to compute a feedback control matrix for the KSE. The adopted formulation allowed all matrix elements to be computed simultaneously. Within a single iteration, a stabilising matrix was computed. Comparisons with standard linear quadratic theory (LQR) showed remarkable similarities (but also some differences) in the computed feedback control kernels. A preconditioned data assimilation algorithm was then derived for state estimation purposes. The preconditioner was again shown to accelerate the rate of convergence significantly. Accurate state estimations were computed for the Lorenz system.