  The following paper discusses the application of a multigrid-in-time scheme
to Least Squares Shadowing (LSS), a novel sensitivity analysis method for
chaotic dynamical systems. While traditional sensitivity analysis methods break
down for chaotic dynamical systems, LSS is able to compute accurate gradients.
Multigrid is used because LSS requires solving a very large Karush-Kuhn-Tucker
(KKT) system constructed from the solution of the dynamical system over the
entire time interval of interest. Several different multigrid-in-time schemes
are examined, and a number of factors were found to heavily influence the
convergence rate of multigrid-in-time for LSS. These include the iterative
method used for the smoother, how the coarse grid system is formed and how the
least squares objective function at the center of LSS is weighted.
