Point selection strategy based on minimizing <italic>GL</italic><sub>2</sub>-discrepancy and its application to multi-dimensional numerical integration

Abstract

An optimal point selection strategy based on minimizing GL 2 -discrepancy, where the effects of assigned probabilities are involved, is proposed in the present paper for multi-dimensional numerical integration and stochastic dynamical response analysis. By doing so the optimal point selection is converted to a nonlinear optimization problem. In the case the coordinates of the points are prescribed, a two-step approach is suggested. The existence and uniqueness of the solution is proved. Through typical numerical examples, the GL 2 -discrepancy of different point sets and the accuracy of the numerical integration are compared, and the effectiveness and advantages of the proposed method is verified. Based on the investigation, the probability density evolution method is employed to implement the nonlinear stochastic dynamic response analysis of a concrete frame structure.