Efficient implementation of high dimensional model representations

Physical models of various phenomena are often represented by a mathematica l model where the output(s) of interest have a multivariate dependence on t he inputs. Frequently, the underlying laws governing this dependence are no t known and one has to interpolate the mathematical model from a finite num ber of output samples. Multivariate approximation is normally viewed as suf fering from the curse of dimensionality as the number of sample points need ed to learn the function to a sufficient accuracy increases exponentially w ith the dimensionality of the function. However, the outputs of most physic al systems are mathematically. well behaved and the scarcity of the data is usually compensated for by additional assumptions on the function (i.e., i mposition of smoothness conditions or confinement to a specific function sp ace). High dimensional model representations (HDMR) are a particular family of representations where each term in the representation reflects the indi vidual or cooperative contributions of the inputs upon the output. The main assumption of this paper is that for most well defined physical systems th e output can be approximated by the sum of these hierarchical functions who se dimensionality is much smaller than the dimensionality of the output. Th is ansatz can dramatically reduce the sampling effort in representing the m ultivariate, function. HDMR has a variety of applications where an efficien t representation of multivariate functions arise with scarce data. The form ulation of HDMR in this paper assumes that the data is randomly scattered t hroughout the domain of the output. Under these conditions and the assumpti ons underlying the HDMR it is argued that the number of samples needed for representation to a given tolerance is invariant to the dimensionality of t he function, thereby providing for a very efficient means to perform high d imensional interpolation. Selected applications of HDMR's are presented fro m sensitivity analysis and time-series analysis.