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 $<J>=\underset{n\to \infty }{\lim }\frac{1}{n}{\sum }_1^{n}J(u_i,s)$ 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 $\frac{d<J>}{ds}$ 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.

Accepted for Publicationin SIAM Journal of Numerical Analysis. The author thanks financial support from AFOSR support under STTR contract FA9550-12-C-0065 through Dr. Fariba Farhoo, and NASA funding through technical monitor Dr. Harold Atkins. The author gratefully acknowledges David Moro and Dr. Si Li for helpful discussion on the proofs

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