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added a small section on RBFs for PDEs and 5 references. Added numbering to the equations as well. |
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\vdots & \vdots & \ddots & \vdots \\
\varphi(\|\mathbf{x}_1 - \mathbf{x}_n\|) & \varphi(\|\mathbf{x}_2 - \mathbf{x}_n\|) & \dots & \varphi(\|\mathbf{x}_n - \mathbf{x}_n\|) \\
\end{bmatrix}, \qquad (1)
</math>
is non-singular.
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* Infinitely Smooth RBFs
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}, \qquad (2)</math>
[[File:Gaussian function shape parameter.png|thumb|right|[[Gaussian function]] for several choices of <math>\varepsilon</math>]]
[[File:Bump function shape.png|thumb|Plot 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}, \qquad (3) </math>
:* [[Inverse quadratic]]:<math display="block">\varphi(r) = \dfrac{1}{1+(\varepsilon r)^2}, \qquad (4) </math>
:* [[Inverse multiquadric]]:<math display="block">\varphi(r) = \dfrac{1}{\sqrt{1 + (\varepsilon r)^2}}, \qquad (5) </math>
* [[Polyharmonic spline]]:<math display="block">\begin{aligned}
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\\
\varphi(r) &= r^k \ln(r),& k&=2,4,6,\dotsc
\end{aligned}, \qquad (6)</math>''*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):<math display="block">\varphi(r) = r^2 \ln(r), \qquad (7)</math>
* Compactly [[Support (mathematics)|Supported]] RBFs
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\exp\left( -\frac{1}{1 - (\varepsilon r)^2}\right) & \mbox{ for } r<\frac{1}{\varepsilon} \\
0 & \mbox{ otherwise}
\end{cases}, \qquad (8)
</math>
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{{main|Radial basis function interpolation}}
Radial basis functions are typically used to build up [[function approximation]]s of the form<math display="block">y(\mathbf{x}) = \sum_{i=1}^N w_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|), \qquad (9)</math>
where the approximating function <math display="inline">y(\mathbf{x})</math> is represented as a sum of <math>N</math> radial basis functions, each associated with a different center <math display="inline">\mathbf{x}_i</math>, and weighted by an appropriate coefficient <math display="inline">w_i.</math> The weights <math display="inline">w_i</math> can be estimated using the matrix methods of [[Weighted least squares|linear least squares]], because the approximating function is ''linear'' in the weights ''<math display="inline">w_i</math>''.
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[[File:Unnormalized radial basis functions.svg|thumb|350px|right|Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at <math display="inline">x_1 = 0.75</math> and <math display="inline">x_2 = 3.25</math>.]]
The sum<math display="block">y(\mathbf{x}) = \sum_{i=1}^N w_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|), \qquad (10)</math>can also be interpreted as a rather simple single-layer type of [[artificial neural network]] called a [[radial basis function network]], with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a [[Compact space|compact]] interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number ''<math display="inline">N</math>'' of radial basis functions is used.
The approximant <math display="inline">y(\mathbf{x})</math> is differentiable with respect to the weights ''<math display="inline">w_i</math>''. The weights could thus be learned using any of the standard iterative methods for neural networks.
Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. {{citation needed|date=February 2019}}
== RBFs for PDEs ==
{{main|Kansa method}}
Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the [[Kansa method]] and was used to solve the elliptic [[Poisson's equation|Poisson equation]] and the linear [[advection-diffusion equation]]. The function values at points <math>\mathbf{x}</math> in the ___domain are approximated by the linear combination of RBFs:
<math display="block">u(\mathbf{x}) = \sum_{i=1}^N \lambda_i \, \varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|),\quad \mathbf{x}\in\R^d \qquad (11)</math>
The derivatives are approximated as such:
<math display="block">\frac{\partial^n u(\textbf{x})}{\partial x^n} = \sum_{i=1}^N \lambda_i \, \frac{\partial^n}{\partial x^n}\varphi(\left\|\mathbf{x} - \mathbf{x}_i\right\|),\quad \mathbf{x}\in\R^d \qquad (12)</math>
where <math>N</math> are the number of points in the discretized ___domain, <math>d</math> the dimension of the ___domain and <math>\lambda</math> the scalar coefficients that are unchanged by the differential operator.<ref>{{Cite journal|last=Kansa|first=E. J.|date=1990-01-01|title=Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations|url=https://www.sciencedirect.com/science/article/pii/089812219090271K|journal=Computers & Mathematics with Applications|language=en|volume=19|issue=8|pages=147–161|doi=10.1016/0898-1221(90)90271-K|issn=0898-1221}}</ref>
Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method<ref>{{Cite journal|last=Tolstykh|first=A. I.|last2=Shirobokov|first2=D. A.|date=2003-12-01|title=On using radial basis functions in a “finite difference mode” with applications to elasticity problems|url=https://doi.org/10.1007/s00466-003-0501-9|journal=Computational Mechanics|language=en|volume=33|issue=1|pages=68–79|doi=10.1007/s00466-003-0501-9|issn=1432-0924}}</ref><ref>{{Cite journal|last=Shu|first=C|last2=Ding|first2=H|last3=Yeo|first3=K. S|date=2003-02-14|title=Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations|url=https://www.sciencedirect.com/science/article/pii/S0045782502006187|journal=Computer Methods in Applied Mechanics and Engineering|language=en|volume=192|issue=7|pages=941–954|doi=10.1016/S0045-7825(02)00618-7|issn=0045-7825}}</ref>, the RBF-QR method<ref>{{Cite journal|last=Fornberg|first=Bengt|last2=Larsson|first2=Elisabeth|last3=Flyer|first3=Natasha|date=2011-01-01|title=Stable Computations with Gaussian Radial Basis Functions|url=https://epubs.siam.org/doi/10.1137/09076756X|journal=SIAM Journal on Scientific Computing|volume=33|issue=2|pages=869–892|doi=10.1137/09076756X|issn=1064-8275}}</ref> and the RBF-PUM method<ref>{{Cite journal|last=Safdari-Vaighani|first=Ali|last2=Heryudono|first2=Alfa|last3=Larsson|first3=Elisabeth|date=2015-08-01|title=A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications|url=https://doi.org/10.1007/s10915-014-9935-9|journal=Journal of Scientific Computing|language=en|volume=64|issue=2|pages=341–367|doi=10.1007/s10915-014-9935-9|issn=1573-7691}}</ref>.
==See also==
* [[Matérn covariance function]]
* [[Radial basis function interpolation]]
*[[Kansa method]]
== References ==
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