Least Squares Shadowing method for sensitivity analysis of differential equations

TL;DR

This paper proves that the Least Squares Shadowing (LSS) method accurately computes sensitivities of ergodic averages in hyperbolic systems as discretization becomes finer and observation time increases.

Contribution

The paper provides a rigorous proof that the LSS algorithm converges to the true derivative of ergodic averages in hyperbolic systems under discretization and time limits.

Findings

LSS algorithm approaches the exact derivative as timestep approaches zero.

Convergence is established when the observation time tends to infinity.

Theoretical validation supports the use of LSS for sensitivity analysis in hyperbolic systems.

Abstract

For a parameterized hyperbolic system $\frac{du}{dt}=f(u,s)$ the derivative of the ergodic average $\langle J \rangle = \lim_{T \to \infty}\frac{1}{T}\int_0^T J(u(t),s)$ to the parameter $s$ can be computed via the Least Squares Shadowing algorithm (LSS). We assume that the sytem is ergodic which means that $\langle J \rangle$ depends only on $s$ (not on the initial condition of the hyperbolic system). After discretizing this continuous system using a fixed timestep, the algorithm solves a constrained least squares problem and, from the solution to this problem, computes the desired derivative $\frac{d\langle J \rangle}{ds}$. The purpose of this paper is to prove that the value given by the LSS algorithm approaches the exact derivative when the discretization timestep goes to $0$ and the timespan used to formulate the least squares problem grows to infinity.