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arXiv · 1905.04715

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

Xiaopeng Luo·Xin Xu·Herschel Rabitz
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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.

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Xiaopeng Luo, Xin Xu, Herschel Rabitz. 2019-05-12. Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems. https://doi.org/10.1007/s10910-019-01031-2

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