Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design

Abstract

Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor product or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for a range of probabilistic analysis problems. In addition, analytic features of the expansions can be exploited for moment estimation and stochastic sensitivity analysis. In this paper, the latest ideas for tailoring these expansion methods to numerical integration approaches will be explored, in which expansion formulations are modified to best synchronize with tensor-product quadrature and Smolyak sparse grids using linear and nonlinear growth rules. The most promising stochastic expansion approaches are then carried forward for use in new approaches for mixed aleatory-epistemic UQ, employing second-order probability approaches, and design under uncertainty, employing bilevel, sequential, and multifidelity approaches.

Sensitivity study of dynamic systems using polynomial chaos

“…It can be calculated by N = 2 (M + 1) [32], or N = (n − 1) (M + 1) [20]. As stated in [20,33], taking more points does not improve the accuracy of the results.…”

“…The first-order sensitivity function can be obtained straightforwardly from (32) with the estimated coefficientsα i (t) (equation (19) or (23)). Thus, the estimated first-order sensitivity functionŜ i (t) of parameter p i is given by:…”

“…When the number of parameters is very large and the number of evaluations of the integral (25) becomes impractical, a solution is to use Smolyak cubature [19,32]. Smolyak formulas are linear combinations of the products expressed in equation (25), which only takes into account products with relatively few integration points.…”

“…The Gauss-Hermite integration of this equation requires N n computations, where n is the number of random variables. As for the number of the integration points, a minimal Gaussian quadrature order of N = d + 1 will be required to obtain a good accuracy in the coefficients, where d is the degree of the polynomials [19,32,29,20].…”

Macromodeling of general linear systems under stochastic variations

“…The PC expansion describes a stochastic process as a suitable summation of orthogonal (polynomial) basis functions with suitable coefficients and gives an analytical representation of the variability of the system with respect to the random variables under consideration [5], [6]. For an extensive reference to PC theory and applications, the reader may consult [1] - [6].…”