Least squares shadowing sensitivity analysis of a modified Kuramoto–Sivashinsky equation

Abstract

Computational methods for sensitivity analysis are invaluable tools for scientists and engineers investigating a wide range of physical phenomena. However, many of these methods fail when applied to chaotic systems, such as the Kuramoto-Sivashinsky (K-S) equation, which models a number of different chaotic systems found in nature. The following paper discusses the application of a new sensitivity analysis method developed by the authors to a modified K-S equation. We find that least squares shadowing sensitivity analysis computes accurate gradients for solutions corresponding to a wide range of system parameters.

Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic Systems and Turbulent fluid flows

“…The cause of the discrepancy between gradients from LSS and curve fitting is discussed by the authors in their analysis of LSS applied to the K-S equation. 14 We found that the systematic error arises from breaks in the assumption of ergodicity. The convergence of LSS to the true gradient value can be proven if the system is ergodic for all initial conditions.…”

“…The behavior exhibited by both equations has similarities to the behavior observed for large scale turbulent fluid flows. 10,14 Because of this, it is important to understand the source of the systematic error in gradient observed for both systems and determine how to mitigate it, so that LSS may be extended to turbulent fluid flows of interest to aerospace engineers. Figure 2 shows gradients computed using LSS and curve fitting for the K-S equation and Lorenz 96 system.…”

“…(BB T + I)ŵ = Bg (16) Note that the matrix in (16) is identical to the one for tangent MSS in (14). This results in an adjoint algorithm similar to the tangent algorithm, wherev andŵ are the adjoint variables:…”

“…LSS has been applied to a number of systems, including the Lorenz 63 system, 13 a modified KuramotoSivashinsky (K-S) equation, 14 and a direct numerical simulation of homogeneous isotropic turbulence in a box. 15 These successes indicate that LSS can enable gradient-based optimization of systems with complicated, turbulent fluid flows.…”

“…Firstly, since (14) is symmetric positive definite, it can be solved with conjugate gradient or other solvers designed for SPD systems. Also, w(t) can be solved without saving v(t), making MSS slightly more memory efficient.…”