Sensitivity Analysis of Chaotic Problems using a Fourier Approximation of the Least-Squares Adjoint

In general, chaotic dynamical systems produce ill-conditioned direct and adjoint sensitivity equations, so these conventional sensitivity analysis methods do not provide useful derivatives for design optimization of chaotic systems. The least-squares shadowing (LSS) method is a recently proposed regularization of the sensitivity equations that leads to useful derivative information; however, the LSS method requires the solution of a second-order boundary-value problem in time that is prohibitively expensive for large-scale simulations. The primary contribution of this work is the observation that the LSS primal and dual solution spaces do not need to be the same, and they can be different dimensions when discretized. For example, we investigate the approximation of the LSS adjoint using a Fourier sine series, and we obtain encouraging results on a model problem based on the Lorenz ordinary differential equation. For the Lorenz problem, the dimensions of the LSS system are reduced by an order of magnitude.