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We develop an algorithm for the equivariant divergence formula of the unstable perturbation of unstable transfer operators, by progressively computing 2u many bounded vectors or covectors on one orbit, where u is the unstable dimension. Combining this with the nonintrusive adjoint shadowing algorithm, we obtain the fast adjoint response algorithm for the linear responses on physical measures of discrete-time uniformly hyperbolic systems. The cost is independent of the number of parameters. We demonstrate our algorithm on a 21-dimensional chaotic system, a system with stochastic noise, and systems with discontinuities. Unlike previous methods, our algorithms are not cursed by dimensionality or the sensitive dependence on initial conditions. [ABSTRACT FROM AUTHOR]

Nonintrusive shadowing algorithms efficiently compute v, the difference between shadowing trajectories, and then use v to compute derivatives of averaged objectives of chaos with respect to parameters of the dynamical system. However, previous proofs of shadowing methods wrongly assume that shadowing trajectories are representative. In contrast, the linear response formula is proved rigorously but is more difficult to compute. We prove that v gives only a part, which is called the shadowing contribution, of the linear response; hence, the other part, the unstable contribution, is the systematic error of shadowing methods. For systems with a small ratio of unstable dimensions, with some further statistical assumptions, we show that the unstable contribution is small. We also briefly describe an algorithm for the unstable contribution, which is simpler to derive but less efficient than the fast linear response algorithm. Moreover, we prove the convergence of the nonintrusive shadowing algorithm, the fastest shadowing algorithm, to v and to the shadowing contribution. [ABSTRACT FROM AUTHOR]