Abstract
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 language | English (US) |
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Pages (from-to) | 1652-1669 |
Number of pages | 18 |
Journal | Journal of Mathematical Chemistry |
Volume | 57 |
Issue number | 6 |
DOIs | |
State | Published - Jun 15 2019 |
All Science Journal Classification (ASJC) codes
- General Chemistry
- Applied Mathematics
Keywords
- Finite difference method
- Hermite-HDMR approximation
- High-dimensional Dirichlet problems
- Meshless method