Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems

Xiaopeng Luo, Xin Xu, Herschel Rabitz

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to 30 show that resulting approximations are of very high quality.

Original languageEnglish (US)
Pages (from-to)1652-1669
Number of pages18
JournalJournal of Mathematical Chemistry
Issue number6
StatePublished - Jun 15 2019

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • Applied Mathematics


  • Finite difference method
  • Hermite-HDMR approximation
  • High-dimensional Dirichlet problems
  • Meshless method


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