Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation
Comparison of Moving Least Squares and RBF+poly for Interpolation and Derivative Approximation
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DOI：
10.1007/s10915-019-01028-8
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发表时间：
2019-08
影响因子：
2.5
通讯作者：
V. Bayona
中科院分区：
文献类型：
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作者：
V. Bayona
多调和样条（PHS）与高次多项式（PHS+poly）的组合最近为径向基函数生成的有限差分近似提供了新的机会。PHS+poly公式依赖于多项式最小二乘拟合来执行局部多项式再现属性，在某种程度上类似于所谓的移动最小二乘（MLS）方法。虽然这两种无网格方法现在越来越多地使用，但还没有进行直接的比较。本研究旨在填补这一空白，重点是分散的数据插值和导数近似。我们首先回顾MLS的方法，并表明，在一些温和的假设PHS+聚可以制定类似。基于启发式的观点和数值模拟，我们然后比较它们的性能在1-D和2-D。一个关键的结果是，如先前发现的PHS+聚，MLS也可以克服边缘振荡（龙格现象），通过简单地增加模板大小为一个固定的多项式次数。然而，这是由加权最小二乘拟合控制的，该拟合对于高多项式次数失败。总体而言，发现PHS+聚乙烯在准确性和稳健性方面表现上级。
The combination of polyharmonic splines (PHS) with high degree polynomials (PHS+poly) has recently opened new opportunities for radial basis function generated finite difference approximations. The PHS+poly formulation, which relies on a polynomial least squares fitting to enforce the local polynomial reproduction property, resembles somehow the so-called moving least squares (MLS) method. Although these two meshfree approaches are increasingly used nowadays, no direct comparison has been done yet. The present study aims to fill this gap, focusing on scattered data interpolation and derivative approximation. We first review the MLS approach and show that under some mild assumptions PHS+poly can be formulated analogously. Based on heuristic perspectives and numerical demonstrations, we then compare their performances in 1-D and 2-D. One key result is that, as previously found for PHS+poly, MLS can also overcome the edge oscillations (Runge’s phenomenon) by simply increasing the stencil size for a fixed polynomial degree. This is, however, controlled by a weighted least squares fitting which fails for high polynomial degrees. Overall, PHS+poly is found to perform superior in terms of accuracy and robustness.