A frequency-domain shadowing approach for sensitivity analysis of chaotic systems

We present a frequency-domain approach for the evaluation of sensitivities of time-averaged quantities of chaotic systems with respect to input parameters. These sensitivities cannot be computed using standard adjoint methods because of the exponential growth of the adjoint variables (due to the presence of positive Lyapunov exponents). The proposed method is based on the well-established least-square shadowing (LSS) approach, that formulates the evaluation of sensitivities as an optimisation problem. Existing formulations of LSS (and its variants) are in the time domain. A reformulation of the LSS method in the frequency (Fourier) space using harmonic balancing is presenred. The resulting system is closed using periodicity. The method is tested on the chaotic Kuramoto-Sivashinsky equation, and the results match with those obtained using the time-domain formulation. However, the storage and computing requirements grow rapidly with the size of the system. To mitigate these requirements, we propose a resolvent-based iterative method that needs much less storage. Application to the Kuramoto-Sivashinsky system gave accurate results with low computational cost. Truncating the large frequencies with small energy content from the harmonic balancing operator did not affect the accuracy of the computed sensitivities. Further details can be found in K. Kantarakias and G. Papadakis (2023), J Comp. Physics, vol. 474, 111757, https://doi.org/10.1016/j.jcp.2022.111757