On the fundamental conjecture of HDMR: a Fourier analysis approach
详细信息 查看全文
- 作者：Xiaopeng Luo ; Xin Xu ; Herschel Rabitz
- 关键词：High dimensional functions ; Multiple Fourier series ; Fourier ; HDMR approximation ; HDMR fundamental conjecture ; Domain decomposition
- 刊名：Journal of Mathematical Chemistry
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：55
- 期：2
- 页码：632-660
- 全文大小：
- 刊物类别：Chemistry and Materials Science
- 刊物主题：Physical Chemistry; Theoretical and Computational Chemistry; Math. Applications in Chemistry;
- 出版者：Springer International Publishing
- ISSN：1572-8897
- 卷排序：55
文摘
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.