Fast adjoint differentiation of chaos via computing unstable perturbations of transfer operators

We devise the fast adjoint response algorithm for the gradient of physical measures (long-time-average statistics) of discrete-time hyperbolic chaos with respect to many system parameters. Its cost is independent of the number of parameters. The algorithm transforms our new theoretical tools, the adjoint shadowing lemma and the equivariant divergence formula, into the form of progressively computing $u$ many bounded vectors on one orbit. Here $u$ is the unstable dimension. We demonstrate our algorithm on an example difficult for previous methods, a system with random noise, and a system of a discontinuous map. We also give a short formal proof of the equivariant divergence formula. Compared to the better-known finite-element method, our algorithm is not cursed by dimensionality of the phase space (typical real-life systems have very high dimensions), since it samples by one orbit. Compared to the ensemble/stochastic method, our algorithm is not cursed by the butterfly effect, since the recursive relations in our algorithm is bounded.