A new multivariate quadrature rule for calculating statistical moments of stochastic response

Abstract

As a derivative-free algorithm, the multivariate quadrature rule has been widely used to calculate statistical moments of the output of a system with uncertain inputs. In this paper, by using Rosenblatt transformation and copula method, the system inputs are related to an independent standard normal random vector U; then, a multivariate polynomial model is introduced to relate the system outputs to U, whereby a set of moment matching equations are derived to define quadrature weights and points. After reviewing the tensor product (TP) based quadrature rule and sparse grid method (SGM), the univariate dimension reduction (UDR) method is reformulated by Kronecker product. Following this routine, a new multivariate quadrature rule is derived by combining a Hadamard matrix based quadrature rule and discrete sine transformation matrix (DSTM). Compared with TP and SGM, the proposed quadrature rule can significantly alleviate the curse of dimensionality, its computational burden increases linearly with respect to the number of uncertain inputs. Besides, the proposed algorithms can match moment matching equations neglected by the UDR method, and thus perform more robustly in the uncertainty quantification problem. Finally, numerical examples are performed to check the proposed quadrature rule.