Stochastic collocation for correlated inputs

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.