Modified Dimension Reduction-Based Polynomial Chaos Expansion for Nonstandard Uncertainty Propagation and Its Application in Reliability Analysis

Jeongeun Son,Yuncheng Du

doi:10.3390/pr9101856

Abstract

This paper presents an algorithm for efficient uncertainty quantification (UQ) in the presence of many uncertainties that follow a nonstandard distribution (e.g., lognormal). Using the polynomial chaos expansion (PCE), the algorithm builds surrogate models of uncertainty as functions of a standard distribution (e.g., Gaussian variables). The key to build these surrogate models is to calculate PCE coefficients of model outputs, which is computationally challenging, especially when dealing with models defined by complex functions (e.g., nonpolynomial terms) under many uncertainties. To address this issue, an algorithm that integrates the PCE with the generalized dimension reduction method (gDRM) is utilized to convert the high-dimensional integrals, required to calculate the PCE coefficients of model predictions, into several lower-dimensional ones that can be rapidly solved with quadrature rules. The accuracy of the algorithm is validated with four examples in structural reliability analysis and compared to other existing techniques, such as Monte Carlo simulations and the least angle regression-based PCE. Our results show our algorithm provides accurate UQ results and is computationally efficient when dealing with many uncertainties, thus laying the foundation to address UQ in complex control systems.

Highlights

Uncertainty, originating from the inherent randomness of a complex system, is common in first-principle models widely used in various engineering problems
Suppose that a stochastic process can be defined as u = K(g), where g = (g1, . . . , gn) ∈ Rn is a Gaussian vector involving n uncertain parameters (n ≥ 1) and K is the function to describe the relationship between a model response u and g
To obtain a polynomial chaos expansion (PCE) expression of u, the first step is to rewrite each parametric uncertainty gi as a function of a Gaussian variable xi as in [20]: p gi = gi(xi) = ∑ gi,k Hk(xi) k=0 where gi,k are the PCE coefficients to estimate the ith parametric uncertainty, which is defined by the ith standard normal distribution, i.e., xi~N (0, 1)

Summary

Introduction

Uncertainty, originating from the inherent randomness of a complex system, is common in first-principle models widely used in various engineering problems. Gn) ∈ Rn is a Gaussian vector involving n uncertain parameters (n ≥ 1) and K is the function to describe the relationship between a model response u and g. It is assumed each parameter in g is independent, i.e., any correlation among uncertain parameters is not considered. K=0 where gi,k are the PCE coefficients to estimate the ith parametric uncertainty, which is defined by the ith standard normal distribution, i.e., xi~N (0, 1) These coefficients gi,k are often assumed to be a given a priori or can be estimated with parameter estimation techniques [40].

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Conclusion