Angxiu Ni,Beijing International Center for Mathematical Research,Peking University

We devise fast algorithms for computing the linear response, which is the parameter-gradient of long-time statistics of hyperbolic chaotic dynamical systems. The algorithm is in the form of progressively computing $u$-many vectors on one orbit, where $u$ is the unstable dimension. So it is much more efficient than previous algorithms (ensemble/stochastic method and finite-element method) for real-life systems with very high dimensional phase spaces.
The fast response algorithm is based on two new formulas. The linear response consists of two parts, a shadowing contribution given by the change of orbits, and an unstable contribution given by the change of conditional measures. For the shadowing contribution, we devise the adjoint shadowing lemma, which extends the conventional backpropagation method to cases with abundant gradient-explosions. For the unstable contribution, we devise the equivariant divergence formula for the unstable perturbation of measure transfer operators.