Stochastic Collocation for Correlated Inputs

Abstract

Abstract. Stochastic Collocation (SC) has been studied and used in different disciplines for Uncertainty Quantification (UQ). The method consists of computing a set of appropriate points, called collocation points, and then using Lagrange interpolation to construct the probability density function (pdf) of the quantity of interest (QoI). The collocation points are usually chosen as Gauss quadrature points, i.e., the roots of orthogonal polynomials with respect to the pdf of the uncertain inputs. If the mathematical model has more than one stochastic parameter, the multidimensional set of points is usually build using the tensor product of the roots of the onedimensional orthogonal polynomials. As a result of that, for multidimensional problems the same set of collocation points is used for both correlated and uncorrelated inputs. In this work, we propose to compute an alternative set of points for correlated inputs. The set will be derived using the orthogonal polynomials for correlated inputs that we developed in a previous work. As these polynomials are not unique, we will obtain multiple sets of collocations points for each input pdf. The aim of this paper is to study the differences between those sets of points and to find and optimal one.

Clustering-Based Collocation for Uncertainty Propagation With Multivariate Dependent Inputs

“…If the input variables are dependent, grids constructed as tensor products of 1-dimensional Gaussian quadrature nodes no longer give rise to a Gaussian cubature rule. In [12], generalization to dependent inputs is approached by constructing sets of polynomials that are orthogonal with respect to general multivariate input distributions, using Gram-Schmidt orthogonalization. The roots of such a set of polynomials can serve as nodes for a Gaussian cubature rule.…”

“…With the approach pursued in [12], the advantages of Gaussian quadrature (in particular, its high degree of exactness) carry over to the multivariate, dependent case. However, one encounters several difficulties with this approach.…”

“…Rather, the resulting set depends on the precise ordering of the monomials that enter the GS procedure. For example, with 2-dimensional inputs and cubic monomials, 24 different sets of nodes can be constructed, as demonstrated in [12]. It is not obvious a priori which of these sets is optimal.…”

Generating Nested Quadrature Rules with Positive Weights based on Arbitrary Sample Sets

“…The applicability of the quadrature rule to complex uncertainty propagation cases is demonstrated by determining the statistics of the flow over an airfoil governed by the Euler equations, including the case of dependent uncertain input parameters. The new quadrature rule significantly outperforms classical sparse grid methods.Other collocation techniques that can be applied to the setting in this work are techniques to consider the collocation problem as a minimization problem of an integration error [18,36], to construct nested rules based on interpolatory Leja sequences [3,21,26], or to apply standard quadrature techniques after decorrelation of the distribution [11,27]. All these approaches provide high order convergence, but require that the input distribution is explicitly known.On the other hand, procedures that directly construct collocation sequences on samples without using the input distribution directly have seen an increase in popularity, possibly due to the recent growth of data sets.…”

“…Other collocation techniques that can be applied to the setting in this work are techniques to consider the collocation problem as a minimization problem of an integration error [18,36], to construct nested rules based on interpolatory Leja sequences [3,21,26], or to apply standard quadrature techniques after decorrelation of the distribution [11,27]. All these approaches provide high order convergence, but require that the input distribution is explicitly known.…”