Fast differentiation of hyperbolic chaos

Understand

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.