On the approximation error in high dimensional model representation

Mathematical models are often described by multivariate functions, which are usually approximated by a sum of lower dimensional functions. A major problem is the approximation error introduced and the factors that affect it. This paper investigates the error of approximating a multivariate function by a sum of lower dimensional functions in the setting of high dimensional model representations. Two kinds of approximations are studied, namely, the approximation based on the ANOVA (analysis of variance) decomposition and the approximation based on the anchored decomposition. We prove new theorems for the expected errors of approximations based on anchored decomposition when the anchor is chosen randomly and establish the relationship of the expected approximation errors with the global sensitivity indices of Sobol'. The expected approximation error give indications on how good or how bad could be the approximation based on anchored decomposition and when the approximation is good or bad. The influence of the anchor on the goodness of approximation is studied. Methods for choosing good anchors are presented.