用于多层级无网格椭圆型偏微分方程求解器的节点子采样
Node Subsampling for Multilevel Meshfree Elliptic PDE Solvers
AI总结：
本文提出了一种针对变密度节点集的新子采样方法及两种节点集质量度量，通过多层级 RBF-FD 迭代求解 Poisson 和 Laplace 方程，实现了线性时间的高阶解和稳健收敛。
摘要：
节点集的子采样在多层级方法、计算机图形学和机器学习等场景中非常有用。在基于均匀网格的节点集上，子采样过程很简单。然而，在密度变化较大的节点集上，通过节点剔除来粗化节点集的过程更为有趣。本文提出了一种针对变密度节点集子采样的新方法。此外，还提出了两种新的节点集质量度量方法，用于确定子采样方法保持初始节点集质量的能力。该新子采样方法在求解 Poisson 和 Laplace 方程的测试问题上进行了演示，通过多层级径向基函数生成的有限差分（RBF-FD）迭代实现。在节点集大小方面，实现了线性时间的高阶解和稳健收敛。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
