The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible bu...The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible but of acceptable accuracy. A generalized L2（GL2）-discrepancy is studied by taking unequal weights for the point set. The extended Koksma-Hlawka inequality is discussed. Thereby, a worst-case error estimate is provided by such defined GL2-discrepancy, whose dosed-form expression is available. The characteristic values of GL2-discrepancy are investigated. An optimal strategy for the selection of the representative point sets with a prescribed cardinal number is proposed by minimizing the GL2-discrepancy. The three typical examples of the multi-dimensional integrals are investigated. The stochastic dynamic response analysis of a nonlinear structure is then studied by incorporating the proposed method into the probability density evolution method. It is shown that the proposed method is advantageous in achieving tradeoffs between the efficiency and accuracy of the exemplified problems. Problems to be further studied are discussed.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.51538010&51261120374)the State Key Laboratory of Disaster Reduction in Civil Engineering(Grant No.SLDRCE14-B-17)the Fundamental Funding for Central Universities
文摘The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible but of acceptable accuracy. A generalized L2（GL2）-discrepancy is studied by taking unequal weights for the point set. The extended Koksma-Hlawka inequality is discussed. Thereby, a worst-case error estimate is provided by such defined GL2-discrepancy, whose dosed-form expression is available. The characteristic values of GL2-discrepancy are investigated. An optimal strategy for the selection of the representative point sets with a prescribed cardinal number is proposed by minimizing the GL2-discrepancy. The three typical examples of the multi-dimensional integrals are investigated. The stochastic dynamic response analysis of a nonlinear structure is then studied by incorporating the proposed method into the probability density evolution method. It is shown that the proposed method is advantageous in achieving tradeoffs between the efficiency and accuracy of the exemplified problems. Problems to be further studied are discussed.
