Sensitivity Analysis of Time-Averaged Quantities of Chaotic Systems

Abstract

Sensitivity analysis of chaotic dynamics in resolved turbulence simulations is still considered a major technical challenge, and existing methods are cost-prohibitive for practical applications. Accurate and efficient sensitivity analysis of such responses is necessary for successful application of design optimization to systems that exhibit chaotic dynamics. This paper presents a fundamentally new approach for sensitivity analysis of time-averaged quantities of chaotic systems. This includes the development of a stabilized time integrator with adaptive time-step control to maintain stability of the sensitivity calculations. Therefore, the approach maintains the initial-value formulation of the problem and avoids costly boundary-value space-time formulation in the least-square shadowing approaches. Both direct- and adjoint-sensitivity analyses approaches are discussed. The method is then applied to the Lorenz system and Kuramoto–Sivashinsky equation, where the sensitivity of time-averaged quantities is computed. The results show that the method provides a robust computational approach with effectively no extra cost other than to compute the sensitivity using a standard implicit time integrator for linear systems.