TY - JOUR
T1 - On the fundamental conjecture of HDMR
T2 - a Fourier analysis approach
AU - Luo, Xiaopeng
AU - Xu, Xin
AU - Rabitz, Herschel Albert
PY - 2017/2/1
Y1 - 2017/2/1
N2 - Although the HDMR decomposition has become an important tool for the understanding of high dimensional functions, the fundamental conjecture underlying its practical utility is still open for theoretical analysis. In this paper, we introduce the HDMR decomposition in conjunction with the Fourier-HDMR approximation leading to the following conclusions: (1) we suggest a type of Fourier-HDMR approximation for certain classes of differentiable functions; (2) utilizing the Fourier-HDMR method, we prove the fundamental conjecture about the dominance of low order terms in the HDMR expansion under relevant conditions, and we also obtain error estimates of the truncated HDMR expansion up to order u; (3) we prove the domain decomposition approximation theorem which shows that the global Fourier-HDMR approximation is not always optimal for a given accuracy order; (4) and finally, a piecewise Fourier-HDMR approach is discussed for high dimensional modeling. These results help to further understand how to efficiently represent the high dimensional functions.
AB - Although the HDMR decomposition has become an important tool for the understanding of high dimensional functions, the fundamental conjecture underlying its practical utility is still open for theoretical analysis. In this paper, we introduce the HDMR decomposition in conjunction with the Fourier-HDMR approximation leading to the following conclusions: (1) we suggest a type of Fourier-HDMR approximation for certain classes of differentiable functions; (2) utilizing the Fourier-HDMR method, we prove the fundamental conjecture about the dominance of low order terms in the HDMR expansion under relevant conditions, and we also obtain error estimates of the truncated HDMR expansion up to order u; (3) we prove the domain decomposition approximation theorem which shows that the global Fourier-HDMR approximation is not always optimal for a given accuracy order; (4) and finally, a piecewise Fourier-HDMR approach is discussed for high dimensional modeling. These results help to further understand how to efficiently represent the high dimensional functions.
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U2 - 10.1007/s10910-016-0701-0
DO - 10.1007/s10910-016-0701-0
M3 - Article
AN - SCOPUS:84991608451
SN - 0259-9791
VL - 55
SP - 632
EP - 660
JO - Journal of Mathematical Chemistry
JF - Journal of Mathematical Chemistry
IS - 2
ER -