论文标题
部分可观测时空混沌系统的无模型预测
MGM: A meshfree geometric multilevel method for systems arising from elliptic equations on point cloud surfaces
论文作者
论文摘要
我们开发了一种新的无网状几何多级（MGM）方法，用于求解由点云代表的表面上离散的椭圆PDE产生的线性系统。该方法使用泊松磁盘采样型技术来使点云和新的网格限制/插值操作员基于多谐波花纹，以在浓缩点云之间传输信息。然后将它们与V-Cycle迭代中的标准平滑和算子粗化方法结合使用。 MGM适用于基于各种局部网格的方法，包括RBF有限差异（RBF-FD）和广义有限差异（GFD），适用于椭圆PDE的离散化。我们使用RBF-FD和GFD测试了MGM作为Krylov子空间方法的独立求解器和预处理程序，并以数值分析收敛速率，效率和缩放率随点云的增加而分析。我们还与代数多机（AMG）方法进行了并排比较，以解决相同的系统。最后，我们通过将其应用于复杂表面上的三个具有挑战性的应用：模式形成，表面谐波和地球距离，进一步证明了MGM的有效性。
We develop a new meshfree geometric multilevel (MGM) method for solving linear systems that arise from discretizing elliptic PDEs on surfaces represented by point clouds. The method uses a Poisson disk sampling-type technique for coarsening the point clouds and new meshfree restriction/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. MGM is applicable to discretizations of elliptic PDEs based on various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We test MGM both as a standalone solver and preconditioner for Krylov subspace methods on several test problems using RBF-FD and GFD, and numerically analyze convergence rates, efficiency, and scaling with increasing point cloud sizes. We also perform a side-by-side comparison to algebraic multigrid (AMG) methods for solving the same systems. Finally, we further demonstrate the effectiveness of MGM by applying it to three challenging applications on complicated surfaces: pattern formation, surface harmonics, and geodesic distance.
