Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds

Chandramoorthy, Nisha; Wang, Qiqi

doi:10.1137/21m1405599

Cited by 7 publications

(19 citation statements)

References 53 publications

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“…The S3 algorithm relies on several recursive formulas in the form of recursive tangent equations. Earlier studies [23,24] proved both analytically and numerically that these recursions converge exponentially fast in discrete hyperbolic systems. We numerically investigate if these results still apply the Lorenz 63 flow.…”

Section: Numerical Example: Lorenz 63mentioning

confidence: 95%

“…Therefore, integration by parts would be possible only if χ belongs to unstable manifolds everywhere in M , which is generally not the case. Motivated by the work of Ruelle [7,8], the authors of [23,10] proposed a new method, called the space-split sensitivity (S3), which regularizes Ruelle's series for systems with one-dimensional unstable subspaces (m = 1). Based on its extension to general hyperbolic maps in [24], we derive and describe a spacesplit approach for chaotic flows with unstable manifolds of arbitrary dimension (m ≥ 1).…”

Section: Ruelle's Formalism and S3mentioning

confidence: 99%

“…Each of them can be approximated through ergodic-averaging of a single (in stable part) or many (in neutral and unstable parts) ingredients. Recent rigorous [23] and computational [24] studies have shown that the rate of convergence of all linear response parts is proportional to 1/ √ N , where N denotes the trajectory length. We highlight the fact that these studies were restricted to hyperbolic systems only.…”

Section: Ruelle's Formalism and S3mentioning

confidence: 99%

“…Based on this idea of regularization of Ruelle's closed-form expression, two conceptually similar methods for linear response emerged in the past two years. Those are the fast linear response algorithm [22] and space-split sensitivity (S3) algorithm [23,24]. The former additionally uses shadowing methods to approximate one of the terms resulting from the perturbation splitting.…”

Section: Introductionmentioning

confidence: 99%

“…The S3 method, on the other hand, does not introduce any approximations except for the ergodic-averaging required for the evaluation of Lebesgue integrals inherited from the original formula. Indeed, the S3 method rigorously converges as a typical Monte Carlo procedure for any uniformly hyperbolic system [23]. Nevertheless, both methods can be summarized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Approximating linear response of physical chaos

Śliwiak¹,

Wang²

2022

Preprint

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Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle's formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring a statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai-Ruelle-Bowen) measure change, which is an integral part of the full linear response, can be completely neglected if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a reduced linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 63, Lorenz 96, and Kuramoto-Sivashinsky.

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Section: Numerical Example: Lorenz 63mentioning

confidence: 95%

Section: Ruelle's Formalism and S3mentioning

confidence: 99%

Section: Ruelle's Formalism and S3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximating linear response of physical chaos

Śliwiak¹,

Wang²

2022

Preprint

Self Cite

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show abstract

Approximating the linear response of physical chaos

Śliwiak

Wang

2022

Nonlinear Dyn

Self Cite

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Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

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Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach

Kantarakias

Papadakis²

2023

Journal of Computational Physics

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Efficient Computation of Linear Response of Chaotic Attractors with One-Dimensional Unstable Manifolds

Cited by 7 publications

References 53 publications

Approximating linear response of physical chaos

Approximating linear response of physical chaos

Approximating the linear response of physical chaos

Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach

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