Probability density adjoint for sensitivity analysis of the Mean of Chaos

TL;DR

This paper introduces the density adjoint method for sensitivity analysis of ergodic chaotic systems, enabling detailed sensitivity computations by solving for the invariant measure on the attractor, demonstrated on maps and Lorenz system.

Contribution

It presents a novel density adjoint approach that computes sensitivities of long-time averages in chaotic systems by focusing on the invariant measure on the attractor.

Findings

Accurately computes sensitivities of chaotic systems.

Provides detailed adjoint distributions.

Has high computational cost.

Abstract

Sensitivity analysis, especially adjoint based sensitivity analysis, is a powerful tool for engineering design which allows for the efficient computation of sensitivities with respect to many parameters. However, these methods break down when used to compute sensitivities of long-time averaged quantities in chaotic dynamical systems. The following paper presents a new method for sensitivity analysis of {\em ergodic} chaotic dynamical systems, the density adjoint method. The method involves solving the governing equations for the system's invariant measure and its adjoint on the system's attractor manifold rather than in phase-space. This new approach is derived for and demonstrated on one-dimensional chaotic maps and the three-dimensional Lorenz system. It is found that the density adjoint computes very finely detailed adjoint distributions and accurate sensitivities, but suffers from large computational costs.