Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos

Ni, Angxiu

doi:10.1137/22m1522383

Cited by 4 publications

(3 citation statements)

References 49 publications

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“…This allows the unstable contribution be also computed with O(u) cost per step. The adjoint theory and algorithm for the unstable contribution of discrete-time systems are in [45,48]. The theory of continuous-time case uses the adjoint shadowing lemma in continuous-time [66].…”

Section: Resultsmentioning

confidence: 99%

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Backpropagation in hyperbolic chaos via adjoint shadowing

2024

Nonlinearity

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To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S ( ω ) is the only bounded inhomogeneous adjoint solution of ω. By (a), S adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), S also expresses the other part of the linear response, the unstable contribution. By (c), S can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.

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Section: Resultsmentioning

confidence: 99%

“…This formula can be computed by 2u recursive relations on an orbit: this is in some sense the ergodic theorem or Monte-Carlo formula for the linear response. The detailed theory for the equivariant divergence formula for discrete-time is in [48]; the discrete-time algorithm and numerical examples are in [45].…”

Section: A Nimentioning

confidence: 99%

Backpropagation in hyperbolic chaos via adjoint shadowing

2024

Nonlinearity

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“…We would also like to remark that the lack of higher-order regularity of statistical quantities pertaining to a smooth dynamical system is also of great import for applications. For example, the lack of differentiability of unstable Jacobians has posed an obstacle for the efficient computation of linear response using Ruelle's formula [29] (see [6,12,20,24] and [3, section 5.3] for background and a broader overview of linear response). It is worth noting that our choice of area-preserving Blaschke maps as perturbations is not well-suited to numerically study this particular problem (due to the perturbations all sharing Lebesgue measure as their relevant invariant measure, thus giving rise to vanishing derivatives).…”

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A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

Bandtlow,

Just,

Slipantschuk

2024

Nonlinearity

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Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

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Equivariant Divergence Formula for Hyperbolic Chaotic Flows

Ni,

Tong

2024

J Stat Phys

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We prove the equivariant divergence formula for axiom A flow attractors. It is a pointwisely-defined and recursive formula for perturbation of SRB measures along center-unstable manifolds. It depends on only the zeroth and first order derivatives of the map, the observable, and the perturbation. Hence, the linear response acquires an ‘ergodic theorem’, which means that it can be sampled by recursively computing a few vectors on one orbit.

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Fast Adjoint Algorithm for Linear Responses of Hyperbolic Chaos

Cited by 4 publications

References 49 publications

Backpropagation in hyperbolic chaos via adjoint shadowing

Backpropagation in hyperbolic chaos via adjoint shadowing

A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps

Equivariant Divergence Formula for Hyperbolic Chaotic Flows

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