A non-intrusive algorithm for sensitivity analysis of chaotic flow simulations

Abstract

We demonstrate a novel algorithm for computing the sensitivity of statistics in chaotic flow simulations to parameter perturbations. The algorithm is non-intrusive but requires exposing an interface. Based on the principle of shadowing in dynamical systems, this algorithm is designed to reduce the e↵ect of the sampling error in computing sensitivity of statistics in chaotic simulations. We compare the e↵ectiveness of this method to that of the conventional finite di↵erence method.

On the Application of Optimal Control Techniques to the Shadowing Approach for Time Averaged Sensitivity Analysis of Chaotic Systems

“…As in the LSS and MSS methods, the time horizon must also be large for accurate sensitivites, Ref. [20]. Adjoint approaches to NILSS have also been developed, Refs.…”

“…[110], NILSS, Refs. [14,20,65,91,92], Periodic Shadowing, Refs. [77,78], and various other approaches, Ref.…”

“…The reason for this is that the Kolmogorov system is spatially two dimensional. The Navier-Stokes equations have been widely investigated for sensitivity analysis of chaotic systems, Refs [14,16,20,21,22,23,32,33,49,90,91,92,93,120,121,127], however, to the author's knowledge sensitivity analysis has not been undertaken on the Kolmogorov system. In this chapter the analysis of the adjoint solutions generated by OCS and NV are compared to MSS to aid in the insight as to what features and structures the control term in the NV method should resemble.…”

Towards Aerodynamic Shape Optimization of Unsteady Turbulent Flows

“…The feasibility of approximating the LSS adjoint with Fourier modes was studied in [15], but this approximation still requires an expensive linear solve, as it alters the problem from a large sparse linear system to a smaller dense linear system. The non-intrusive least squares shadowing (NILSS) method [16][17][18] seeks to address the computational cost of the LSS method, while still computing useful gradient information. While it shows promise, NILSS requires a potentially large (> 100) number of linearized partial differential equation (PDE) solves, one for each positive Lyapunov exponent.…”