Convergence of the least squares shadowing method for computing derivative of ergodic averages

TL;DR

This paper proves that the least squares shadowing method for computing derivatives of ergodic averages in hyperbolic systems converges to the true derivative as the problem size increases.

Contribution

It provides a rigorous proof of convergence for the least squares shadowing method in hyperbolic dynamical systems.

Findings

Convergence of the least squares shadowing method to the true derivative.

Mathematical proof of the method's validity for large problem sizes.

Applicability to parameterized hyperbolic systems.

Abstract

For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\ < J\ > = \underset{n\rightarrow\infty}{\lim} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares sensitivity method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $d\ < J\ > \over ds$ from the solution. This paper proves that as the size of the least squares problem approaches infinity, the computed approximation converges to the true derivative.