Reproducing kernel Hilbert space interpolation methods as a paradigm of high dimensional model representations: Application to multidimensional potential energy surface construction

Abstract

A generic high dimensional model representation (HDMR) method is presented for approximating multivariate functions in terms of functions of fewer variables and for going beyond the tensor-product formulation. Within the framework of reproducing kernel Hilbert space (RKHS) interpolation techniques, an HDMR is formulated for constructing global potential energy surfaces. The HDMR tools in conjunction with a successive multilevel decomposition technique provide efficient and accurate procedures for reducing a multidimensional interpolation problem to smaller, independent subproblems. It is shown that, when compared to the conventional tensor-product approach, the RKHS–HDMR methods can accurately produce smooth potential energy surfaces over dynamically relevant, nonrectangular regions using far fewer ab initio data points. Numerical results are given for a reduced two-level RKHS–HDMR of the C(1D)+H2 reactive system. The proposed RKHS–HDMR is intimately related to Gordon’s blending-function methods for multivariate interpolation and approximation. The general findings in the paper and the successful illustration provide a foundation for further applications of the techniques.