Radial basis function: Difference between revisions

Revision as of 06:53, 5 March 2021

Revision as of 06:55, 5 March 2021

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These radial basis functions are from <math>C^\infty(\mathbb{R})</math> and are strictly [[Positive-definite function|positive definite functions]]<ref>{{cite book |last1=Fasshauer |first1=Gregory E. |title=Meshfree Approximation Methods with MATLAB |date=2007 |publisher=World Scientific Publishing Co. Pte. Ltd. |___location=Singapore |isbn=9789812706331 |pages=37–45}}</ref> that require tuning a shape parameter <math>\varepsilon</math>

:* [[Gaussian function|Gaussian]]:<math display="block">\varphi(r) = e^{-(\varepsilon r)^2}</math>

[[File:Gaussian function shape parameter.png|thumb|right|A [[Gaussian function]] for several choices of <math>\varepsilon</math>]]

[[File:Bump function shape.png|thumb|A plotPlot of the scaled [[Bump function|bump function]] with several choices of <math>\varepsilon</math>]]

:* [[Multiquadric]]:<math display="block">\varphi(r) = \sqrt{1 + (\varepsilon r)^2} </math>