In this talk we discuss how to compute derivatives of long-time-averaged objectives with respect to multiple system parameters in chaotic systems, via the recently developed non-intrusive least-squares adjoint shadowing (NILSAS) algorithm.
First we review how to compute such derivatives via comparing the base trajectory and a shadowing trajectory, which is a new trajectory with perturbed parameter and perturbed initial condition, yet always lies close to the base trajectory. Then we review how to compute such shadowing trajectory via a `non-intrusive' minimization problem on the unstable subspace. Then we show our recent work on defining and proving the unique existence of adjoint shadowing directions. Then we develop the NILSAS algorithm, whose cost is independent of number of parameters, and its implementation requires only minor modifications to existing adjoint solvers. Finally, we show an application, by Chaitanya Talnikar, of NILSAS on a weakly turbulent flow over a three-dimensional cylinder at Re=1100, where the cost of NILSAS is similar to simulating the flow problem.