Reproducing kernel technique for high dimensional model representations (HDMR)

An easy and effective approach is proposed to estimate the arbitrary l order HDMR approximations for complex high dimensional physical systems on the basis of the reproducing kernel Hilbert space (RKHS). With the help of Fourier transform and Dirac delta function, the corresponding explicit reproducing kernel K(x,y) is first constructed to approximate the HDMR approximations by a linear combination of K(x,y). Then the computation of the l order HDMR approximations can be given in the form of solving a system of linear equations. It can be strictly proved that this linear system is just another equivalent definition of the lth order HDMR approximations by using the corresponding reproducing kernel. And the numerical examples provide a practical evidence for the rationality and effectiveness of the proposed approach.