  In this computational paper, we perform sensitivity analysis of long-time (or
ensemble) averages in the chaotic regime using the shadowing algorithm. We
introduce automatic differentiation to eliminate the tangent/adjoint equation
solvers used in the shadowing algorithm. In a gradient-based optimization, we
use the computed shadowing sensitivity to minimize different long-time averaged
functionals of a chaotic time-delayed system by optimal parameter selection. In
combined state and parameter estimation for data assimilation, we use the
computed sensitivity to predict the optimal trajectory given information from a
model and data from measurements beyond the predictability time. The algorithms
are applied to a thermoacoustic model. Because the computational framework is
rather general, the techniques presented in this paper may be used for
sensitivity analysis of ensemble averages, parameter optimization and data
assimilation of other chaotic problems, where shadowing methods are applicable.
