关于径向基函数计算的一个注记
A note on radial basis function computing
摘要：
本说明有三个目的，涉及我们的径向基函数（RBF）方法的最新进展。首先，我们将介绍一个新的格式，采用边界结方法（BKM）的非线性对流扩散问题。它强调，新的计划直接导致在一个线性的BKM制定的非线性问题，通过使用响应点相关的径向基函数，它可以解决任何线性求解器。然后，我们只需要解决一个单一的非线性代数方程的期望未知数在一些内部节点的兴趣。数值结果表明，这种非线性BKM策略具有较高的精度和效率。其次，我们扩展了距离函数的概念，包括时空距离函数和变变换距离函数。最后，我们证明，如果节点是对称放置，RBF系数矩阵具有中心对称或斜中心对称结构。这样的矩阵的因式分解功能导致在RBF计算工作量的相当大的减少。在一般情况下，提出了一种简单的方法来减少RBF插值矩阵的病态。
英文摘要：
This note carries three purposes involving our latest advances on the radial basis function (RBF) approach. First, we will introduce a new scheme employing the boundary knot method (BKM) to nonlinear convection-diffusion problem. It is stressed that the new scheme directly results in a linear BKM formulation of nonlinear problems by using response point-dependent RBFs, which can be solved by any linear solver. Then we only need to solve a single nonlinear algebraic equation for desirable unknown at some inner node of interest. The numerical results demonstrate high accuracy and efficiency of this nonlinear BKM strategy. Second, we extend the concepts of distance function, which include time-space and variable transformation distance functions. Finally, we demonstrate that if the nodes are symmetrically placed, the RBF coefficient matrices have either centrosymmetric or skew centrosymmetric structures. The factorization features of such matrices lead to a considerable reduction in the RBF computing effort. A simple approach is also presented to reduce the ill-conditioning of RBF interpolation matrices in general cases.
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