Shadowing Methods for Forward and Adjoint Sensitivity Analysis of Chaotic Systems | FS

In this post, we dig into sensitivity analysis of chaotic systems. Chaotic systems are dynamical, deterministic systems that are extremely sensitive to small changes in the initial state or the system parameters. Specifically, the dependence of a chaotic system on its initial conditions is well known as the “butterfly effect”. Chaotic models are encountered in various fields ranging from simple examples such as the double pendulum to highly complicated fluid or climate models.

Sensitivity analysis methods have proven to be very powerful for solving inverse problems such as parameter estimation or optimal control1 2 3. However, conventional sensitivity analysis methods may fail in chaotic systems due to the ill-conditioning of the initial value problem. Sophisticated methods, such as least squares shadowing4 (LSS) or non-intrusive least squares shadowing5 (NILSS) have been developed in the last decade. Essentially, these methods transform the initial value problem to a well conditioned optimization problem – the least squares shadowing problem. In this second part of my GSoC project, I implemented the LSS and the NILSS method within the DiffEqSensitivity.jl package.

The objective for LSS and NILSS is a long-time average quantity. More precisely, we define the instantaneous objective by $g(u,p)$, where $u$ is the state and $p$ is the parameter of the differential equation. Then, the objective is obtained by averaging $g$ over an infinitely long trajectory:

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