Practical uncertainty quantification analysis involving statistically dependent random variables

This article presents a practical refinement of generalized polynomial chaos expansion for uncertainty quantification under dependent input random variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For uncertainty quantification analysis under dependent random variables, two regression methods, comprising existing standard least-squares and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of generalized polynomial chaos expansion for the very first time. In contrast to the existing regression devoted so far to the classical polynomial chaos expansion, no tensor-product structure is required or enforced. The partitioned D-MORPH regression is applicable to either an underdetermined or overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Numerical results obtained for Gaussian and non-Gaussian probability measures with rectangular or non-rectangular domains point to highly accurate orthonormal polynomials produced by the three-step algorithm. More significantly, the generalized polynomial chaos approximations of mathematical functions and stochastic responses from solid-mechanics problems, in tandem with the partitioned D-MORPH regression, provide excellent estimates of the second-moment properties and reliability from only hundreds of function evaluations or finite element analyses.