Node Subsampling for Multilevel Meshfree Elliptic PDE Solvers

TL;DR

This paper introduces a new subsampling method for variable density node sets used in meshfree PDE solvers, improving efficiency and solution quality in multilevel RBF-FD methods.

Contribution

A novel subsampling technique for high-density variation node sets and new quality measures, enabling efficient multilevel PDE solutions.

Findings

Achieves high-order solutions with robust convergence

Operates in linear time relative to node set size

Demonstrates effectiveness on Poisson and Laplace problems

Abstract

Subsampling of node sets is useful in contexts such as multilevel methods, computer graphics, and machine learning. On uniform grid-based node sets, the process of subsampling is simple. However, on node sets with high density variation, the process of coarsening a node set through node elimination is more interesting. A novel method for the subsampling of variable density node sets is presented here. Additionally, two novel node set quality measures are presented to determine the ability of a subsampling method to preserve the quality of an initial node set. The new subsampling method is demonstrated on the test problems of solving the Poisson and Laplace equations by multilevel radial basis function-generated finite differences (RBF-FD) iterations. High-order solutions with robust convergence are achieved in linear time with respect to node set size.