An Anisotropic Sparse Grid Numerical Integration-Based Statistic Moment Estimation Method

Statistic moment estimation is a paramount task in design optimization under uncertainty. In the existing isotropic sparse grid-based statistic moment estimation method, each dimension is considered to be of equal importance and collocated with the same number of integration points. Therefore, "curse of dimensionality" is still very serious. To address this issue, a new statistic moment estimation approach is developed by employing the anisotropic sparse grid technique from the numerical integration point of view in this paper. A new rule allowing different accuracy levels to each dimension according to their varying importance is proposed to determine the eligible multi-index combinations. Integration points can be collocated more reasonably, and more points can be removed from unimportant dimensions to the important ones. The effectiveness of the proposed approach is demonstrated through comparative studies on several mathematical examples and a practical engineering example. It is observed that compared with the existing method, the proposed approach can further mitigate "curse of dimensionality" and significantly improve the accuracy of statistic moment estimation with even less computational cost.