Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems
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 show that resulting approximations are of very high quality.
Cite
@article{arxiv.1905.04715,
title = {Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems},
author = {Xiaopeng Luo and Xin Xu and Herschel Rabitz},
journal= {arXiv preprint arXiv:1905.04715},
year = {2019}
}