On the fundamental conjecture of HDMR: a Fourier analysis approach

Xiaopeng Luo, Xin Xu, Herschel Albert Rabitz

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)632-660
Number of pages29
JournalJournal of Mathematical Chemistry
Volume55
Issue number2
DOIs
StatePublished - Feb 1 2017

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Chemistry(all)

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