  Computational methods for sensitivity analysis are invaluable tools for
aerodynamics research and engineering design. However, traditional sensitivity
analysis methods break down when applied to long-time averaged quantities in
turbulent fluid flow fields, specifically those obtained using high-fidelity
turbulence simulations. This is because of a number of dynamical properties of
turbulent and chaotic fluid flows, most importantly high sensitivity of the
initial value problem, popularly known as the "butterfly effect".
  The recently developed least squares shadowing (LSS) method avoids the issues
encountered by traditional sensitivity analysis methods by approximating the
"shadow trajectory" in phase space, avoiding the high sensitivity of the
initial value problem. The following paper discusses how the least squares
problem associated with LSS is solved. Two methods are presented and are
demonstrated on a simulation of homogeneous isotropic turbulence and the
Kuramoto-Sivashinsky (KS) equation, a 4th order chaotic partial differential
equation. We find that while LSS computes fairly accurate gradients, faster,
more efficient linear solvers are needed to apply both LSS methods presented in
this paper to larger simulations.
