Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach

Abstract

ElsevierChen Charpentier, BM.; Cortés, J.; Licea Salazar, JA.; Romero, J.; Roselló Ferragud, MD.; Santonja, F.; Villanueva Micó, RJ. (2015). Constructing adaptive generalized polynomial chaos method to measure the uncertainty in continuous models: A computational approach. AbstractDue to errors in measurements and inherent variability in the quantities of interest, models based on random differential equations give more realistic results than their deterministic counterpart. The generalized polynomial chaos (gPC) is a powerful technique used to approximate the solution of these equations when the random inputs follow standard probability distributions. But in many cases these random inputs do not have a standard probability distribution. In this paper, we present a step-by-step constructive methodology to implement directly a useful version of adaptive gPC for arbitrary distributions, extending the applicability of the gPC. The paper mainly focuses on the computational aspects, on the implementation of the method and on the creation of a useful software tool. This tool allows the user to easily change the types of distributions and the order of the expansions, and to study their the usefulness of the method are included.

Computational uncertainty quantification for random time‐discrete epidemiological models using adaptive gPC

“…Instead of this, these polynomials are constructed directly from the random parameter inputs, by a Gram‐Schmidt orthonormalization procedure (if the random inputs are independent) or by using polynomial canonical bases (if the random inputs are not independent). The numerical experiments from previous studies showed that adaptive gPC provides reliable results for small basis orders of the solution process. This allows us to obtain the main statistical moments associated with the solution process: expectation, variance, etc.…”

“…It must be remarked that, although the theoretical support of the model coefficients is contained in [0,1], a distribution that may leave [0,1] but with a negligible probability (for instance, a normal or gamma distributions with mean in [0,1] and a very small variance) could be possible in practice. The goal is to quantify the uncertainty for S ( m ), E ( m ), I ( m ), and R ( m ) using adaptive gPC …”

“…We will work in the Hilbert space of random variables whose variance exists, (L 2 (Ω),⟨,⟩), where the inner product ⟨,⟩ is defined as , being the expectation operator. Denote ζ 1 = a , ζ 2 = b , ζ 3 = c , and ζ 4 = μ , and ζ = ( ζ 1 , ζ 2 , ζ 3 , ζ 4 ) ⊤ (⊤ is the transpose operator), so that the notation matches with previous studies . Let be the canonical basis of the polynomials in ζ i up to degree p , 1 ≤ i ≤ 4.…”

“…Recently, in previous studies, the authors modified gPC to an adaptive gPC computational approach. In the adaptive gPC approach, the orthogonal polynomials do not belong to the Askey‐Wiener scheme.…”

Stochastic Collocation for Correlated Inputs

“…In this method to find the polynomial coefficients either Galerkin projection or Spectral projection is used, which results in a coupled system of deterministic equations. Since the first version of this method for Gaussian random variables appeared, until now several generalizations and improvements of the PCE technique have been published: [2,3,20,18], etc. So nowadays, due to increased relevance of UQ during the last years, a wide variety of techniques to compute the uncertainties in the model have been applied, for instance: Karhunen-Loève decomposition [11], gradientbased methods [24], sparse grids [7], perturbation methods [6] based on local Taylor series expansions, Bayesian methods [10] etc.…”