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An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

机译：混合高斯-三次核的改进的径向基-伪谱方法

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摘要

While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.
机译：尽管伪光谱（PS）方法具有很高的准确性，但在几何灵活性方面往往受到严重限制。全局径向基函数的应用克服了这一问题，但是以有问题的条件为代价（1）以它们最精确的平坦基函数形式，以及（2）当问题规模扩大到具有实际意义时。本研究考虑了通过使用结合了三次样条和高斯核的混合径向基函数来改善这两个问题的策略。使用全局粒子群优化来选择控制混合RBF中的高斯和立方核的参数。所提出的方法已经用径向基-伪谱方法进行了泊松，亥姆霍兹和输运方程数值近似的测试。观察到，所提出的方法显着减少了RBF-PS方法中的不适状况问题，同时，它为非常小的形状参数保留了稳定性和准确性。发现改进算法中系数矩阵的特征值谱即使在很大的自由度下也能保持稳定，这与伪光谱方法获得的谱图相似。而且，数值实验表明，混合核的性能明显优于纯高斯核和纯立方核。

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