Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

Śliwiak, Adam A.; Chandramoorthy, Nisha; Wang, Qiqi

doi:10.1016/j.cnsns.2021.105906

Cited by 13 publications

(21 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The computation of these two quantities is the actual price for the regularization of the original Lebesgue integrals. The latter is known in the literature as the SRB density gradient [25,29,24]. It reflects a relative measure change along an unstable manifold and its value is thus independent from its corresponding quotient measure.…”

Section: Computation Of the Unstable Contributionmentioning

confidence: 99%

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

Sliwiak,

Wang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Accurate approximations of the change of system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the sensitivity of chaotic dynamical systems, which enables a better understanding of chaotic phenomena. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting regularizing Ruelle's original expression together with measure-based parameterization of the expanding subspace. We use these tools to rigorously derive trajectory-following recursive relations that exponentially converge, and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics. Thanks to the regularization and lack of simplifying assumptions on the behavior of the system, our method is immune to the common problems of other popular systems such as the exploding tangent solutions and unphysicality of shadowing directions. We provide a ready-to-use algorithm, analyze its complexity, and demonstrate several numerical examples of sensitivity computation of physically-inspired low-dimensional systems.

show abstract

Section: Computation Of the Unstable Contributionmentioning

confidence: 99%

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

Sliwiak,

Wang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Such an assumption reduces the FDT linear response operator to a simple time autocorrelation function, which dramatically facilitates the sensitivity computation for the cost of limited applicability. The density gradient can also be used as an reliable indicator of the differentiability of statistical quantities [30] in chaotic systems. In particular, the slope of the distribution tail of g have been shown to be strictly associated with the existence of parametric derivatives of statistics.…”

mentioning

confidence: 99%

“…In case of simple one-dimensional maps, one can derive an exponentially convergent recursion for g using the measure preservation property [29]. The same formula can be inferred using the fact the SRB density is an eigenfunction of the Frobenius-Perron operator with eigenvalue 1 [30]. The authors of [9] propose an ergodic-averaging algorithm for self-derivatives (i.e., directional derivatives along one-dimensional expanding directions) of covariant Lyapunov vectors (CLVs) corresponding to the only positive Lyapunov exponent, which are tangent to unstable manifolds at any point on the attractor.…”

mentioning

confidence: 99%

“…The authors of [9] propose an ergodic-averaging algorithm for self-derivatives (i.e., directional derivatives along one-dimensional expanding directions) of covariant Lyapunov vectors (CLVs) corresponding to the only positive Lyapunov exponent, which are tangent to unstable manifolds at any point on the attractor. Using the chain rule on smooth manifolds, one can show g depends on the self-derivative of CLV at the previous time step, and this relation is governed by a second-order tangent equation [9,30]. In a recent work, Ni [24] proposes an algorithm for divergence on the unstable manifold using approximate shadowing coordinates instead of the full basis of the expanding subspace, as opposed to the S3 method.…”

mentioning

confidence: 99%

“…Since M Df k • q dµ = − M f k g dµ, we can thus alternatively apply Monte Carlo to the RHS that involves the SRB density gradient g [10,30,29],…”

mentioning

confidence: 99%

See 2 more Smart Citations

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

Sliwiak,

Wang

2021

Preprint

Self Cite

View full text Add to dashboard Cite

SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectorydriven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.

show abstract

Approximating the linear response of physical chaos

Śliwiak

Wang

2022

Nonlinear Dyn

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

Cited by 13 publications

References 45 publications

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

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

A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

Approximating the linear response of physical chaos

Contact Info

Product

Resources

About