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Sigmoid Functions, Multiscale Resolution of Singularities, and hp-Mesh Refinement
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DOI：10.1137/23M1556629.png)
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Abstract
En 中文
In this short, conceptual paper we observe that closely related mathematics applies in four contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth refinement for solution of PDEs, and (4) double exponential (DE) and generalized Gauss quadrature. The relationships start from the change of variables s = log(x), and they suggest possibilities for new analyses and new methods in several areas. Concerning (2) and (3), we show that both problems feature the same effect of ``linear tapering near the singularity---of clustered poles in rational approximation and of polynomial orders in hpmesh refinement. Concerning (4), we note that the tapering effect appears here too, and that the change of variables interpretation sheds new light on why the DE and generalized Gauss methods are effective at integrating arbitrary singularities.
Keywords:
rational approximation
sigmoid function
logistic function
radial basis function
hp-mesh refinement
DE quadrature
generalized Gauss quadrature
Journal
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6.1
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888
Citations:
1.2W
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