Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS)

This paper develops the Non-Intrusive Least Squares Shadowing (NILSS) method, which computes the sensitivity for long-time averaged objectives in chaotic dynamical systems. In NILSS, we represent a tangent solution by a linear combination of one inhomogeneous tangent solution and several homogeneous tangent solutions. Next, we solve a least squares problem using this representation; thus, the resulting solution can be used for computing sensitivities. NILSS is easy to implement with existing solvers. In addition, for chaotic systems with many degrees of freedom but few unstable modes, NILSS has a low computational cost. NILSS is applied to two chaotic PDE systems: the Lorenz 63 system and a CFD simulation of flow over a backward-facing step. In both cases, the sensitivities computed by NILSS reflect the trends in the long-time averaged objectives of dynamical systems.