Radial basis function: Difference between revisions

Revision as of 22:10, 28 August 2022

Revision as of 15:29, 3 January 2023

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The derivatives are approximated as such:

{{NumBlk||<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 </math>|{{EquationRef|12}}}}

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|doi-access=free}}</ref>

Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method,<ref>{{Cite journal|lastlast1=Tolstykh|firstfirst1=A. I.|last2=Shirobokov|first2=D. A.|date=2003-12-01|title=On using radial basis functions in a “finite"finite difference mode”mode" with applications to elasticity problems|url=https://link.springer.com/article/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 |s2cid=121511032 | issn=1432-0924}}</ref><ref>{{Cite journal|lastlast1=Shu|firstfirst1=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|lastlast1=Fornberg|firstfirst1=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|lastlast1=Safdari-Vaighani | firstfirst1=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://link.springer.com/article/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 | s2cid=254691757 |issn=1573-7691}}</ref>

==See also==