Feasibility Analysis of Ensemble Sensitivity Computation in Turbulent Flows

Abstract

In chaotic systems, such as turbulent flows, the solutions to tangent and adjoint equations exhibit an unbounded growth in their norms. This behavior renders the instantaneous tangent and adjoint solutions unusable for sensitivity analysis. The Lea-Allen-Haine ensemble sensitivity (ES) estimates provide a way of computing meaningful sensitivities in chaotic systems by utilizing tangent/adjoint solutions over short trajectories. In this paper, we analyze the feasibility of ES computations under optimistic mathematical assumptions on the flow dynamics. Furthermore, we estimate upper bounds on the rate of convergence of the ES method in numerical simulations of turbulent flow. Even at the optimistic upper bound, the ES method is computationally intractable in each of the numerical examples considered. Nomenclature ES Ensemble Sensitivity τ trajectory length for ES computation N number of i.i.d samples used in ES computation θ τ, N ES estimator Superscripts T, A and FD stand for tangent, adjoint and finite difference respectively u a d-dimensional state (or phase) vector used to represent a primal states scalar parameter u(t, s, u 0 ) primal state vector at time t with initial state u 0u(t, s), u(t) this notation is used to represent the primal state when the

Space-split algorithm for sensitivity analysis of discrete chaotic systems with unstable manifolds of arbitrary dimension

“…Instead of generating a long trajectory, Eyink et al [7] proposed computing sensitivities of several truncated-in-time trajectories and taking the average of the partial results. While this approach does not suffer from the butterfly effect and is proven to work in different real-world chaotic systems [8], large variances of the partial estimates make the ensemble methods prohibitively expensive even for medium-sized models. Yet another popular family of methods derives from the shadowing lemma [9] which, under the assumption of uniform hyperbolicity, guarantees the existence of a shadowing trajectory that lies withing a small distance to the reference solution for a long (but finite) time.…”

Sensitivity computation of statistically stationary quantities in turbulent flows

“…In contrast, the variance of the integrand in the original form (Eq.17), grad 0 (J n ) · X u 0 , exhibits unbounded growth with n . Thus, an N-term ergodic average of Eq.17 shows a (much) slower convergence than the 1/ √ N convergence predicted by the central limit theorem (see [21] for an analysis of ensemble estimates of the gradient term).…”

“…An early approach, known as ensemble sensitivity computation, due to Lea et al [1], proposed to take the sample average of short time sensitivities computed using these linearized perturbation equations. However, the ensemble sensitivity approach is computationally intractable, as some studies have shown [2,21]. The more recent shadowing-based approaches [3-5, 22, 23] tackle this problem by numerically computing the shadowing perturbation solution, a carefully constructed tangent or an adjoint solution that remains bounded over a long time window.…”

From limited observations to the state of turbulence: Fundamental difficulties of flow reconstruction

“…The associated computational cost is always on the order of twice that of calculating the cost function, regardless of the size of the control vector. However, for chaotic systems the accuracy of the gradient deteriorates for long integration time horizons [27,28]. Therefore, the implementation of adjoint methods is restricted to short assimilation windows that can be concatenated to span long periods of observations [20,29].…”