Experiments with an adaptive multicut-HDMR map generation for slowly varying continuous multivariate functions

摘要

In many areas of science an technology there is a need for efficient methods of approximating multivariate functions. Methods of this kind often suffer from the exponential growth (known as the curse of dimensionality) of the computational costs, with the increasing number of independent variables. The approximation called cut-HDMR (High Dimensional Model Representation), elaborated in recent years, avoids this problem, but is valid only locally. Its extensions, called multicut-HDMR approximations, are applicable to larger function domains, but are less developed. Numerical experiments performed in this study reveal advantages and weaknesses of three multicut-HDMR variants known from the literature, and two new variants defined by us. The experiments focus on an adaptive, error-driven generation of approximants to slowly varying continuous functions. Accuracy, convergence rate, computational times, and amounts of data needed, are compared. Three most satisfactory variants are identified. Two of them average multiple cut-HDMR maps using weights dependent on distances to cut subspaces. In the third one the grid of cut points splits the function domain into hyperrectangles, and a single cut-HDMR map having the smallest errors represents the approximant in every hyperrectangle.