Convergence of the Least Squares Shadowing Method for Computing Derivative of Ergodic Averages

Abstract

For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\langle J\rangle = \lim_{n\rightarrow\infty} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares shadowing method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $\frac{d\langle J\rangle}{ds}$ 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.