Fast differentiation of hyperbolic chaos

Abstract

We derive and prove the `fast response' formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic chaotic systems. The expression is pointwisely defined so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. It has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm's cost is solving $u$, the unstable dimension, many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods.

References in corpus (13)

• Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NILSS)
• Convergence of the least squares shadowing method for computing derivative of ergodic averages
• Fluctuations, Response, and Resonances in a Simple Atmospheric Model
• Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic Dynamical Systems
• Linear response, or else
• Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow
• Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)
• Periodic Shadowing Sensitivity Analysis of Chaotic Systems
• Sensitivity analysis on chaotic dynamical systems by Finite Difference Non-Intrusive Least Squares Shadowing (FD-NILSS)
• Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems
• A computable realization of Ruelle's formula for linear response of statistics in chaotic systems
• A Preconditioned Multiple Shooting Shadowing Algorithm for the Sensitivity Analysis of Chaotic Systems
• Adjoint shadowing directions in hyperbolic systems for sensitivity analysis

Cited by in corpus (3)

• Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems
• Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems
• Approximating linear response by nonintrusive shadowing algorithms