Development of Adaptive RBF-HDMR Model for Approximating High Dimensional Problems

Abstract

Modeling or approximating high dimensional, computationally-expensive, black-box problems faces an exponentially increasing difficulty, the “curse-of-dimensionality”. This paper proposes a new form of high-dimensional model representation (HDMR) by integrating the radial basis function (RBF). The developed model, called RBF-HDMR, naturally explores and exploits the linearity/nonlinearity and correlation relationships among variables of the underlying function that is unknown or computationally expensive. This work also derives a lemma that supports the divide-and-conquer and adaptive modeling strategy of RBF-HDMR. RBF-HDMR circumvents or alleviates the “curse-of-dimensionality” by means of its explicit hierarchical structure, adaptive modeling strategy tailored to inherent variable relation, sample reuse, and a divide-and-conquer space-filling sampling algorithm. Multiple mathematical examples of a wide scope of dimensionalities are given to illustrate the modeling principle, procedure, efficiency, and accuracy of RBF-HDMR.