Efficient sensitivity analysis method for chaotic dynamical systems

Liao, Hua

doi:10.1016/j.jcp.2016.02.016

Cited by 12 publications

(15 citation statements)

References 23 publications

(40 reference statements)

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“…The paper proposes a new approach to analyse the sensitivity of algorithms for estimating the status of indicators of harmful information observed in noise. An analysis of current works aimed at the sensitivity of complex processes and systems, for example [8][9][10][11][12], from the point of view of the technical aspects of novelty, emphasizes the motivation of the paper and confirms most of the statements made in this paper.…”

Section: Introductionsupporting

confidence: 70%

Analysis of the Sensitivity of Algorithms for Assessing the Harmful Information Indicators in the Interests of Cyber-Physical Security

Kotenko

Parashchuk

2019

Electronics

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The secure functioning of cyber-physical systems depends on the presence and amount of harmful (unwanted and malicious) information in its digital network content. The functioning of cyber-physical systems is carried out in non-stationary conditions and in conditions of continuous exposures. This leads to the uncertainty of indicators (parameters, features) of harmful information that must be assessed in the analytical processing of digital network content. The paper proposes an approach to analyse the sensitivity of algorithms for estimating the status of indicators of harmful information observed in noise. This approach allows one to consider possible errors in the estimation accuracy. It gives the possibility to identify the allowable range of changes in the parameters of the digital network content of cyber-physical systems, within which the requirements for the assessment reliability are met. This, in turn, makes a significant contribution to the effectiveness of harmful information detection and counteraction against it. Accounting for a priori uncertainty of the indicators under various influences is advisable to carry out on the basis of expressions for the sensitivity coefficients (functions) described in the paper.

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Section: Introductionsupporting

confidence: 70%

Analysis of the Sensitivity of Algorithms for Assessing the Harmful Information Indicators in the Interests of Cyber-Physical Security

Kotenko

Parashchuk

2019

Electronics

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“…It can be observed that the arithmetic average of the periodic shadowing sensitivity converges to a value around J T dρ » 1.017 as T is increased. Different variants of LSS produce similar values gradients [41,24]. The sensitivity calculated from UPOs, which is not affected by the shadowing error E T 1 also converges, on average, to such a value.…”

Section: Sensitivity With Respect To Perturbations To ρmentioning

confidence: 76%

“…for some constant c ą 0 that might depend on the frequency of the zeros and the average slope of gpxq near them. With the change of variable R " 1{r, we obtain P pgpxq ă 1{Rq » c{R, r " R. This last appendix considers the spectral properties of the matrix BpT, x 0 q for the aero-elastic oscillator previously used as a test bed for shadowing methods [41,24]. The dynamics of the oscillator are defined by the second order nonlinear differential equation G¨: zptq`D¨9 zptq`pK 1`K2 Qq¨zptq`K 3¨z 3 ptq " 0, (F.1)…”

Section: (D2)mentioning

confidence: 99%

Periodic shadowing sensitivity analysis of chaotic systems

Lasagna

Sharma

Meyers

2019

Journal of Computational Physics

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The sensitivity of long-time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction, the uniformly-bounded-in-time solution of the sensitivity equations. Although its existence is formally guaranteed for certain systems, methods to determine it are hardly available. One practical approach is the Least-Squares Shadowing (LSS) algorithm (Q Wang, SIAM J Numer Anal 52, 156, 2014), whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm. Here, we present an alternative, potentially simpler shadowing-based algorithm, termed periodic shadowing. The key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time. We show that this is not only justifiable when the reference trajectory is itself periodic, but also possible and effective for chaotic trajectories. Our error analysis shows that periodic shadowing has the same convergence rates as LSS when the time span T is increased: the sensitivity error first decays as 1{T and then, asymptotically as 1{?T . We demonstrate the approach on the Lorenz equations, and also show that, as T tends to infinity, periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits (D Lasagna, SIAM J Appl Dyn Syst 17, 1, 2018) for which there is no shadowing error. Finally, finite-difference approximations of the sensitivity are also examined, and we show that subtle non-hyperbolicity features of the Lorenz system introduce a small, yet systematic, bias.

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“…It should also be noted that we are considering a specific form of the LSS sensitivity method without time dilation and its associated weighting parameters. 17 Although this may have an effect on the accuracy of the LSS adjoint, our proposed methods remain applicable to the LSS method with the time dilation term included.…”

Section: A Least-squares Shadowing Adjointmentioning

confidence: 99%

“…, 1, 1/2) holds the quadrature weights corresponding to the trapezoid rule. Again, since the LSS method removes the initial condition from the linearized problem, we remove the first block row from A to give A in (17). To be explicit, A is defined as…”

Section: Discretization Of the Lss Adjointmentioning

confidence: 99%

Sensitivity Analysis of Chaotic Problems using a Fourier Approximation of the Least-Squares Adjoint

Ashley

Hicken

2016

17th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference

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In general, chaotic dynamical systems produce ill-conditioned direct and adjoint sensitivity equations, so these conventional sensitivity analysis methods do not provide useful derivatives for design optimization of chaotic systems. The least-squares shadowing (LSS) method is a recently proposed regularization of the sensitivity equations that leads to useful derivative information; however, the LSS method requires the solution of a second-order boundary-value problem in time that is prohibitively expensive for large-scale simulations. The primary contribution of this work is the observation that the LSS primal and dual solution spaces do not need to be the same, and they can be different dimensions when discretized. For example, we investigate the approximation of the LSS adjoint using a Fourier sine series, and we obtain encouraging results on a model problem based on the Lorenz ordinary differential equation. For the Lorenz problem, the dimensions of the LSS system are reduced by an order of magnitude.

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Efficient sensitivity analysis method for chaotic dynamical systems

Cited by 12 publications

References 23 publications

Analysis of the Sensitivity of Algorithms for Assessing the Harmful Information Indicators in the Interests of Cyber-Physical Security

Analysis of the Sensitivity of Algorithms for Assessing the Harmful Information Indicators in the Interests of Cyber-Physical Security

Periodic shadowing sensitivity analysis of chaotic systems

Sensitivity Analysis of Chaotic Problems using a Fourier Approximation of the Least-Squares Adjoint

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