For a parameterized hyperbolic system du dt = f (u, s) the derivative of the ergodic average J = lim T →∞ 1 T T 0 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 J 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 d J 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.The differential equation is assumed to be uniformly hyperbolic (details in section 3). We are also given a C 1 cost function J(u, s) : U × R → R and assume that the system is ergodic, i.e., the infinite time average :depends on s but does not depend on the initial condition u(0). The differentiability of J with respect to s has been proven by Ruelle [1]. Obtaining an estimation of d J ds is crucial in many computational and engineering problems. Indeed, many applications involve simulations of nonlinear dynamical systems that exhibit a chaotic behavior. For instance, chaos can be encountered in the following fields : climate and weather prediction [2], turbulent combustion simulation [3], nuclear reactor physics [4], plasma dynamics in fusion [5] and multi-body problems [6]. The quantities of interest are often time averages or expected values of some cost function J. Estimating the derivative of J is particularly valuable in:• Numerical optimization. The derivative of J with respect to a design parameter s is used by gradient-based algorithms in order to efficiently optimize the design parameters in high dimensional design spaces (see [7]). • Uncertainty quantification. The derivative of J with respect to a parameter s gives a useful information for assessing the error and uncertainty in the computed J (see [8]).

This paper develops a new variant of the Least Squares Shadowing (LSS) method: NonIntrusive LSS (NI-LSS). Comparing to previous LSS algorithm, this new variant is easier to implement on top of existing tangent/adjoint solvers. Furthermore, for chaotic systems with large degrees of freedom but low dimensional strange attractor, this new variant reduces the computation cost by orders of magnitude. NI-LSS is based on the idea of solving the minimization problem through projection, transforming the optimization arguments to the coefficients of a family of unstable homogeneous solutions, hence the final optimization problem is much smaller. In this paper NI-LSS is applied to two chaotic PDE systems: the Lorenz 63 system, and an airfoil with free-flutter flap. The results show that NI-LSS computes the correct derivative with a lower cost than LSS. Nomenclature J(u, s) quantity of interest valued at each time step f long time average of f , f = lim T →∞ 1 T t end t0 f dt s parameter for the dynamical system m dimension of u, i.e. u ∈ R m m us dimension of unstable subspace, or number of unstable directions n number of time segment t 0 start time of NI-LSS t end end time of NI-LSS T total time for NI-LSS, T = t end − t 0 ∆T length for each time segment, ∆T = t i − t i−1 Subscript i index for time segment, i = 1, . . . , n j index for columns in W , j = 1, . . . , m us

This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be more efficient and simpler to implement, especially for PDEs.

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