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

## TL;DR

This paper introduces a meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems, achieving high accuracy and stability with fewer nodes even in 30 dimensions.

## Contribution

It develops a novel meshless Hermite-HDMR approach with error estimates and smoothing, enabling efficient high-dimensional PDE solutions.

## Key findings

- High-order convergence demonstrated in experiments
- Method maintains accuracy with fewer nodes
- Effective in dimensions up to 30

## 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.