Radial basis function: Difference between revisions

Revision as of 18:36, 23 May 2025

Revision as of 14:54, 26 June 2025

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| [[Inverse multiquadric]]:

{{NumBlk||<math display="block">\varphi(r) = \dfrac{1}{\sqrt{1 + (\varepsilon r)^2}}, </math>|{{EquationRef|5}}}}

| RTH<ref>{{cite journal |last1=Heidari |first1=Mohammad |last2=Mohammadi |first2=Maryam |last3=De Marchi |first3=Stefano |title=A shape preserving quasi-interpolation operator based on a new transcendental RBF |journal=Dolomites Research Notes on Approximation |volume=14 |issue=1 |pages=56–73 |year=2021 |doi=10.14658/PUPJ-DRNA-2021-1-6}}</ref>:

{{NumBlk||<math display="block">\varphi(r) = r \tanh(\varepsilon r), </math>|{{EquationRef|6}}}}

}}

| [[Polyharmonic spline]]:

{{NumBlk||<math display="block">\begin{aligned}

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\\

\varphi(r) &= r^k \ln(r),& k&=2,4,6,\dotsc

\end{aligned} </math>|{{EquationRef|67}}}}

''*For even-degree polyharmonic splines'' <math>(k = 2,4,6,\dotsc)</math>'', to avoid numerical problems at <math>r = 0</math> where <math>\ln(0) = -\infty</math>, the computational implementation is often written as <math>\varphi(r) = r^{k-1}\ln(r^r)</math>.''{{citation needed|date=May 2021}}

| [[Thin plate spline]] (a special polyharmonic spline):

{{NumBlk||<math display="block">\varphi(r) = r^2 \ln(r), </math>|{{EquationRef|78}}}}

| Compactly [[Support (mathematics)|Supported]] RBFs

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0 & \text{ otherwise}

\end{cases},

</math>|{{EquationRef|89}}}}

}}

}}