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 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]).

Periodic shadowing sensitivity analysis of chaotic systems

“…for all t, leading to the "canonical" shadowing direction, as defined in Ref. [10]. If the shadowing direction were known, the sensitivity for the finite-time trajectory xpt; x 0 , pq defined over the time span t P r0, T s could be calculated as J T,S dp px 0 , pq "…”

“…In Ref. [10] it is proven by exchange of limits that the finite-time sensitivity (27) converges to the infinite-time sensitivity J 8 dp , defined by the limit (4), as T Ñ 8. For ergodic, mixing dynamical systems, the central limit theorem dictates the average rate of convergence.…”

“…Providing such boundary conditions directly not only results in a method that is potentially simpler, but it sufficient to obtain bounded (periodic) solutions almost always, resulting in accurate gradients. The paper includes a detailed error analysis section, where we shown that the proposed method has the same asymptotic convergence rates of LSS [40,10], and where we derive asymptotic statistical distribution of the sensitivity error.…”

Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NI-LSS)

“…The objective function J is the square of the lift coefficient, C 2 L , and we are interested in its long time average J. The averaging window for the objective function, is defined on time t ∈ [2,27]. For each design parameter, the objective function is computed as the mean of 4 different runs with random initial conditions.…”

“…24 The case we consider in this paper is a continuous system governed by an ODE, and the corresponding tangent LSS theorem was proved by Chater. 27 Here we use a simpler yet equivalent form summarized by Blonigan. 25 It states that, under ergodicity and hyperbolicity assumptions,…”