Least Squares Shadowing Method for Sensitivity Analysis of Differential Equations

Abstract

For a parameterized hyperbolic system du dt = f(u; s) the derivative of the ergodic average hJi = limT→∞ 1 T R T 0 J(u(t); s) to the parameter s can be computed via the least squares shadowing (LSS) algorithm. We assume that the system is ergodic, which means that hJi depends only on s (not on the initial condition of the hyperbolic system). The algorithm solves a constrained least squares problem and, from the solution to this problem, computes the desired derivative dhJi ds . The purpose of this paper is to prove that the value given by the LSS algorithm approaches the exact derivative when the timespan used to formulate the least squares problem grows to infinity. It then illustrates the convergence result through a numerical example.