Abstract
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).
Highlights
Over the past few decades, modern computational methods have been very successful in predicting the evolution of systems of great complexity
Uncertainty is introduced through s which is modeled by a vector of m random independent variables ξ = (ξ 1, . . . , ξ m ), where each ξ i follows a given probability density function (PDF) wi (ξ i ), defined in the domain Ei
The generalized Polynomial Chaos (gPC) is an established method in propagating and evaluating uncertainties, yet the accuracy of the intrusive version deteriorates when applied to general unsteady systems
Summary
Over the past few decades, modern computational methods have been very successful in predicting the evolution of systems of great complexity. Other approaches, based on the spectral representation of uncertain quantities, have been developed These techniques are referred to as Polynomial Chaos Expansion (PCE) methods. The iPC approach is not feasible for conducting UQ because the accuracy of the spectral representation of the state variables starts to deteriorate after the system starts to evolve from the initial condition For this reason, we used the niPC approach and focused on uncertainties of time-averaged quantities and their sensitivities with respect to input parameters. It is shown that the non-intrusive approach to the gPC is suitable for UQ of time-averages of chaotic systems and their sensitivities, with respect to system parameters 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]
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