Improving Point Selection in Cubature by a New Discrepancy

Reasonable point set selection is of paramount importance to the accuracy of highdimensional integrals that will be encountered in various disciplines. In the present paper, to improve the point selection and to overcome the computational complexity of evaluating classical discrepancies, the concept of extended F-discrepancy (EF-discrepancy) and generalized F-discrepancy (GF-discrepancy) of a point set is introduced and justified by comparative studies with other existing discrepancies. Meanwhile, the extensions of Koksma-Hlawka inequality for EF-discrepancy are proved and a conjecture for GF-discrepancy is put forward and discussed. This GF-discrepancy is then employed as the objective function when selecting the optimal rotation angles in the rotation transform of the quasi-symmetric point method (Q-SPM). Meanwhile, it is also proved that the rotation transform will keep the degree of algebraic accuracy. A genetic algorithm is adopted to solve the optimization problem. Several numerical examples are elaborated, demonstrating that the GF-discrepancy is a reasonable index in judging the goodness of a point set, and that the optimal rotation of Q-SPM will greatly improve the accuracy of stochastic analysis of nonlinear structures. The proposed GF-discrepancy and the resulting rotational Q-SPM point sets could be applied directly to other problems of uncertainty quantification. Problems to be further studied are discussed.