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In this computational paper, we perform sensitivity analysis of long-time (orensemble) averages in the chaotic regime using the shadowing algorithm. Weintroduce automatic differentiation to eliminate the tangent/adjoint equationsolvers used in the shadowing algorithm. In a gradient-based optimization, weuse the computed shadowing sensitivity to minimize different long-time averagedfunctionals of a chaotic time-delayed system by optimal parameter selection. Incombined state and parameter estimation for data assimilation, we use thecomputed sensitivity to predict the optimal trajectory given information from amodel and data from measurements beyond the predictability time. The algorithmsare applied to a thermoacoustic model. Because the computational framework israther general, the techniques presented in this paper may be used forsensitivity analysis of ensemble averages, parameter optimization and dataassimilation of other chaotic problems, where shadowing methods are applicable.

Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity