Adaptive Orthonormal Basis Functions for High Dimensional Metamodeling With Existing Sample Points

Abstract

High Dimensional Model Representation (HDMR) is a tool for generating an approximation of an input-output model for a multivariate function. It can be used to model a black-box function for metamodel-based optimization. Recently the authors’ team has developed a radial basis function based HDMR (RBF-HDMR) model that can efficiently model a high dimensional black-box function and, moreover, to uncover inner variable structures of the black-box function. This approach, however, requests a complete new, although optimized, set of sample points, as dictated by the methodology, while in engineering design practice one often has many existing sample data. How to utilize the existing data to efficiently construct a HDMR model is the focus of this paper. We first identify the Random-Sampling HDMR (RS-HDMR), which uses orthonormal basis functions as HDMR component functions and existing sample points can be used to calculate the coefficients of the basis functions. One of the important issues related to the RS-HDMR is that in theory the basis functions are obtained based on the continuous integrations related to the orthonormality conditions. In practice, however, the integrations are approximated by Monte Carlo summation and thus the basis functions may not satisfy the orthonormality conditions. In this paper, we propose new and adaptive orthonormal basis functions with respect to a given set of sample points for RS-HDMR approximation. RS-HDMR models are built for different test functions using the standard and new adaptive basis functions for different number of sample points. The relative errors for both models are calculated and compared. The results show that the models that are built using the new basis functions are more accurate.