Generalized Anisotropic Sparse Grid Integrals Based on Adaptive High Dimension Model Representation for Moment Estimation

Estimation of statistical moments of structural response effectively and accurately is still one of the main topics for analysis of random systems. For the sake of reducing function evaluations to alleviate curse of dimensionality and improve the accuracy in moment estimation, a generalized anisotropic sparse grid integral based on the adaptive high dimensional model representation and appropriate reference variables is proposed. By introducing different transformations, a system with general variables is transformed into the one with appropriate independent reference variables and a generalized anisotropic sparse grid integral will be presented for moment estimation. Then, anisotropic sparse grid collections should be divided into several mutually exclusive sub-collections and the generalized anisotropic SGI can be remodeled in a new form. With this new form, the adaptive high dimensional model representation can be introduced for function evaluations conveniently, which can be regarded as an alternative way to evaluate functions precisely and efficiently without any approximation. Finally, several examples are presented to demonstrate the effectiveness and accuracy of the proposed method.