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.

翻译：节点子采样在多层级方法、计算机图形学及机器学习等领域具有重要应用价值。对于均匀网格节点集，子采样过程相对简单；但当节点集存在高密度变化时，通过节点消除实现粗化则更具挑战性。本文提出了一种适用于可变密度节点集的新型子采样方法，同时引入两种创新的节点集质量度量指标，用于评估子采样方法保留初始节点集质量的能力。通过求解泊松方程和拉普拉斯方程的多层级径向基函数生成有限差分（RBF-FD）迭代测试问题，验证了该新子采样方法的有效性。数值实验表明，该方法能够实现高阶精度解与稳健收敛性，且计算复杂度与节点规模呈线性关系。