Approximating the linear response of physical chaos

Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle's formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai-Ruelle-Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto-Sivashinsky.

Automated summary

This paper investigates the computation problem of parametric derivatives in the presence of chaos. Based on Ruelle's formalism, it proposes a method to simplify the linear response algorithm in higher-dimensional systems featuring statistical homogeneity. The study shows that the contribution of the SRB measure gradient can be negligible if the objective function is appropriately aligned with unstable manifolds. Several numerical examples support these conclusions and demonstrate the use and performance of the simplified linear response algorithm.