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

Kantarakias, Kyriakos Dimitrios; Shawki, Karim; Papadakis, George

doi:10.1103/physreve.101.022223

Cited by 12 publications

(15 citation statements)

References 67 publications

(110 reference statements)

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“…A comparison with finite difference (FD) data is also presented. For dJ1 dc , there is a small bias, which has also been observed in the time-domain formulation of the method [19,20,42]; for dJ2 dc the matching with FD is very good.…”

Section: Application To the Kuramoto-sivashinsky Equationsupporting

confidence: 52%

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

Kantarakias¹,

Papadakis²

2022

Preprint

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We present a frequency-domain method for computing the sensitivities of timeaveraged quantities of chaotic systems with respect to input parameters. Such sensitivities cannot be computed by conventional adjoint analysis tools, because the presence of positive Lyapunov exponents leads to exponential growth of the adjoint variables. The proposed method is based on the least-square shadowing (LSS) approach [1], that formulates the evaluation of sensitivities as an optimisation problem, thereby avoiding the exponential growth of the solution. However, all existing formulations of LSS (and its variants) are in the time domain and the computational cost scales with the number of positive Lyapunov exponents.In the present paper, we reformulate the LSS method in the Fourier space using harmonic balancing. The new method is tested on the Kuramoto-Sivashinski system and the results match with those obtained using the standard timedomain formulation. Although the cost of the direct solution is independent of the number of positive Lyapunov exponents, storage and computing requirements grow rapidly with the size of the system. To mitigate these requirements, we propose a resolvent-based iterative approach that needs much less storage.Application to the Kuramoto-Sivashinski system gave accurate results with very low computational cost. The method is applicable to large systems and paves the way for application of the resolvent-based shadowing approach to turbulent flows. Further work is needed to assess its performance and scalability.

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Section: Application To the Kuramoto-sivashinsky Equationsupporting

confidence: 52%

“…Variants of the original LSS method include the Multiple Shooting Shadowing (MSS) [18,19,20] and the non-intrusive Least Squares Shadowing (NILSS), [21,22]. Both methods can be applied to large systems, but their computational cost scales with the number of PLEs.…”

Section: Introductionmentioning

confidence: 99%

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

Kantarakias¹,

Papadakis²

2022

Preprint

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“…Information on the latter, i.e., UQ of sensitivities, is important in robust design applications, where the expectation of a QoI is minimized under uncertainty. The present work follows on from previous work of the authors in this area [23].…”

Section: Introductionmentioning

confidence: 97%

Application of Generalized Polynomial Chaos for Quantification of Uncertainties of Time Averages and Their Sensitivities in Chaotic Systems

Kantarakias¹,

Papadakis²

2020

Algorithms

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In this paper, we consider the effect of stochastic uncertainties on non-linear systems with chaotic behavior. More specifically, we quantify the effect of parametric uncertainties to time-averaged quantities and their sensitivities. Sampling methods for Uncertainty Quantification (UQ), such as the Monte–Carlo (MC), are very costly, while traditional methods for sensitivity analysis, such as the adjoint, fail in chaotic systems. In this work, we employ the non-intrusive generalized Polynomial Chaos (gPC) for UQ, coupled with the Multiple-Shooting Shadowing (MSS) algorithm for sensitivity analysis of chaotic systems. It is shown that the gPC, coupled with MSS, is an appropriate method for conducting UQ in chaotic systems and produces results that match well with those from MC and Finite-Differences (FD).

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“…Evaluating a polynomial representation requires much less computational resources compared to the evaluation of an actual model, which makes the sampling of the parameter space far less computationally expensive [11]. When PCE is used to quantify uncertainty in a smooth system with finite variance, it provides exponential convergence at relatively low computational costs [12], and therefore PCE is the most commonly used UQ propagation technique when dealing with smooth processes [13].…”

Section: Introductionmentioning

confidence: 99%

“…This has resulted in authors attempting to make PCE resistant to chaos, nonlinearity and the resulting nonsmoothness. To this end, Kantarakias et al [13] coupled PCE with the multiple shooting shadowing [23] method and applied it to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. It was found that at the quadrature integration points of PCE, the resulting regularization lead to accurate results that matched well with the Monte Carlo simulations at significantly lower computational costs.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification of Time-Dependent Quantities in a System With Adjustable Level of Smoothness

Legkovskis

Thomas

Auinger

2022

Journal of Verification, Validation and Uncertainty Quantification

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We summarise the results of a computational study involved with Uncertainty Quantification (UQ) in a benchmark turbulent burner flame simulation. UQ analysis of this simulation enables one to analyse the convergence performance of one of the most widely-used uncertainty propagation techniques, Polynomial Chaos Expansion (PCE) at varying levels of system smoothness. This is possible because in the burner flame simulations, the smoothness of the time-dependent temperature, which is the study's QoI is found to evolve with the flame development state. This analysis is deemed important as it is known that PCE cannot accurately surrogate non-smooth QoIs and thus perform convergent UQ. While this restriction is known and gets accounted for, there is no understanding whether there is a quantifiable scaling relationship between the PCE's convergence metrics and the level of QoI's smoothness. It is found that the level of QoI-smoothness can be quantified by its standard deviation allowing to observe the effect of QoI's level of smoothness on the PCE's convergence performance. It is found that for our flow scenario, there exists a power-law relationship between a comparative parameter, defined to measure the PCE's convergence performance relative to Monte Carlo sampling, and the QoI's standard deviation, which allows us to make a more weighted decision on the choice of the uncertainty propagation technique.

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

Cited by 12 publications

References 67 publications

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

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

Application of Generalized Polynomial Chaos for Quantification of Uncertainties of Time Averages and Their Sensitivities in Chaotic Systems

Uncertainty Quantification of Time-Dependent Quantities in a System With Adjustable Level of Smoothness

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