Adjoint sensitivity analysis of chaotic systems using cumulant truncation

Craske, John

doi:10.1016/j.chaos.2018.12.024

Cited by 11 publications

(6 citation statements)

References 49 publications

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“…In a series of papers, Wang and co-workers have introduced the idea of least-squares shadowing (LSS) to address this problem [24][25][26][27][28]. Other approaches have also been proposed [29,30]. The LSS method is based on the shadowing lemma which is applicable for ergodic and uniformly hyperbolic systems [31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

2020

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We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE).

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Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

2020

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“…where standard parameters σ = 10, β = 8/3 and ρ = 28 are used throughout. As in other sensitivity studies on the Lorenz equations [10,15,39,43,55,56], we consider the sensitivity of the period average of the observable J(t) = u 3 (t) with respect to perturbations of ρ. Numerical integration of chaotic trajectories is performed using a classical fourth-order Runge-Kutta method with ∆t = 0.005.…”

Section: Numerical Resultsmentioning

confidence: 99%

Sensitivity of long periodic orbits of chaotic systems

Lasagna

2020

Phys. Rev. E

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The properties of long, numerically-determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two-order of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

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“…where standard parameters σ = 10, β = 8/3 and ρ = 28 are used throughout. As in other sensitivity studies on the Lorenz equations 9,14,42,50,59,60 , we consider the sensitivity of the period average of the observable J(t) = u 3 (t) with respect to perturbations of ρ. Numerical integration of chaotic trajectories is performed using a classical fourth-order Runge-Kutta method with ∆t = 0.005.…”

Section: Numerical Resultsmentioning

confidence: 99%

Sensitivity of long periodic orbits of chaotic systems

Lasagna

2019

Preprint

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The relationship between the stability of unstable periodic orbits of dissipative chaotic systems and the sensitivity of period averages to parameter perturbations is clarified. Using Floquet theory, we show that period averages vary most rapidly when structural perturbations of the equations of motion are well expressed by adjoint Floquet eigenfunctions associated to Floquet exponents of small magnitude, along invariant subspaces closest to marginality. Building on this analysis, we then construct a large inventory of periodic orbits of the Lorenz equations and of the Kuramoto-Sivashinsky system in a minimal-domain configuration, focusing on long periodic orbits spanning a large fraction of the chaotic attractor. We examine how the statistical distributions of Floquet exponents, period averages and their sensitivities to parameter perturbations vary as the period T increases. We show that, at least for these two systems, the distribution of these quantities converges to a Dirac delta function as T → ∞. We observe that the limiting value of period averages and Floquet exponents tend to the same asymptotic value obtained on long chaotic trajectories. Conversely, the limiting value of the sensitivity is not consistent with the response of long-time averages to finiteamplitude parameter perturbations, due to the lack of a linear response for non-hyperbolic dynamics.

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Adjoint sensitivity analysis of chaotic systems using cumulant truncation

Cited by 11 publications

References 49 publications

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

Sensitivity of long periodic orbits of chaotic systems

Sensitivity of long periodic orbits of chaotic systems

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