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

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).

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

“…gPC methods have been successful in predicting the statistics of the QoI in many applications, such as fluid dynamics, mechanics, space, medicine, see for example [3,5,[9][10][11]. Applications of gPC to chaotic systems have also appeared in the literature [12][13][14][15][16].…”