Approximating linear response by nonintrusive shadowing algorithms

Nonintrusive shadowing algorithms efficiently compute $v$, the difference between shadowing trajectories, 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, 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.