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

Abstract

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

Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow

“…With high cost, LSS has been successfully applied in two-dimensional (2D) CFD problems (?). The non-intrusive least-squares shadowing (NILSS) method (Ni et al 2016;Ni & Wang 2017) reformulates LSS by constraining computation to the unstable subspace.…”

Adjoint sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Adjoint Shadowing (NILSAS)

“…Hence we can allow its coefficient be O(1) without jeopardizing the boundedness property we used earlier. In fact, the adjoint NILSS in [24] lacks exactly this constraint on the neutral adjoint CLV.…”

A non-intrusive algorithm for sensitivity analysis of chaotic flow simulations

“…A less intrusive algorithm has recently been developed. 10 This paper demonstrates its utility on a chaotic, eddy-resolving turbulent flow simulation.…”