径向基函数的一些近期进展
Some recent advances on the RBF
AI总结：
本文综述了径向基函数（RBF）在数值偏微分方程中的近期进展，包括核RBF、误差界、插值矩阵结构、边界节点法、修正Kansa方法及最小二乘配点方案，这些方法均具有无网格、对称、谱收敛等优点。
摘要：
本文旨在综述我们近期关于径向基函数（RBF）及其在数值偏微分方程中应用的工作。我们引入了包含一般预小波和尺度正交小波RBF的核RBF。我们还猜想了一个与维度无关的RBF误差界。指出了在对称采样节点下RBF插值矩阵的中心对称结构。另一方面，我们通过非奇异一般解和对偶互易原理引入了边界节点法，并通过多重互易原理引入了边界粒子法。利用格林积分，我们开发了一种称为修正Kansa方法的域型Hermite RBF方案，该方案显著减少了边界附近的计算误差。为了规避吉布斯现象，我们提出了最小二乘RBF配点方案。上述所有离散方案都是无网格、对称、谱收敛、免积分且数学上非常简单的。数值验证也简要给出。
英文摘要：
This paper aims to survey our recent work relating to the radial basis function (RBF) and its applications to numerical PDEs. We introduced the kernel RBF involving general pre-wavelets and scale-orthogonal wavelets RBF. A dimension-independent RBF error bound was also conjectured. The centrosymmetric structure of RBF interpolation matrix under symmetric sample knots was pointed out. On the other hand, we introduced the boundary knot method via nonsingular general solution and dual reciprocity principle and the boundary particle method via multiple reciprocity principle. By using the Green integral we developed a domain-type Hermite RBF scheme called the modified Kansa method, which significantly reduces calculation errors around boundary. To circumvent the Gibbs phenomenon, the least square RBF collocation scheme was presented. All above discretization schemes are meshfree, symmetric, spectral convergent, integration-free and mathematically very simple. The numerical validations are also briefly presented.
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