  We develop the NILSAS algorithm, which performs adjoint sensitivity analysis
of chaotic systems via computing the adjoint shadowing direction. NILSAS
constrains its minimization to the adjoint unstable subspace, and can be
implemented with little modification to existing adjoint solvers. The
computational cost of NILSAS is independent of the number of parameters. We
demonstrate NILSAS on the Lorenz 63 system and a weakly turbulent
three-dimensional flow over a cylinder.
