Approximating linear response of physical chaos

TL;DR

This paper simplifies the computation of parametric derivatives in chaotic systems by neglecting certain measure changes under specific conditions, demonstrated through numerical models like Lorenz and Kuramoto-Sivashinsky.

Contribution

It introduces a reduced linear response algorithm that neglects SRB measure change contributions in higher-dimensional chaotic systems with statistical homogeneity.

Findings

Simplified linear response algorithm performs well in numerical experiments.

Neglecting SRB measure change is valid under certain alignment conditions.

Applicable to various physical chaotic models like Lorenz and Kuramoto-Sivashinsky.

Abstract

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 a statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai-Ruelle-Bowen) measure change, which is an integral part of the full linear response, can be completely neglected 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 reduced linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 63, Lorenz 96, and Kuramoto-Sivashinsky.