  It is well-known that linearized perturbation methods for sensitivity
analysis, such as tangent or adjoint equation-based, finite difference and
automatic differentiation are not suitable for turbulent flows. The reason is
that turbulent flows exhibit chaotic dynamics, leading to the norm of an
infinitesimal perturbation to the state growing exponentially in time. As a
result, these conventional methods cannot be used to compute the derivatives of
long-time averaged quantities to control or design inputs. The ensemble-based
approaches and shadowing-based approaches to circumvent the problems of the
conventional methods in chaotic systems, also suffer from computational
impracticality and lack of consistency guarantees, respectively. We introduce
the space-split sensitivity, or the S3 algorithm, which is a Monte-Carlo
approach to the chaotic sensitivity computation problem. In this work, we
derive the S3 algorithm under simplifying assumptions on the dynamics and
present a numerical validation on a low-dimensional example of chaos.
