Javascript is required
[1] Kansa, E.J., Multiquadrics - a scattered data approximation scheme with applications tocomputational fluid dynamics I: surface approximations and partial derivative estimates.Computers & Mathematics with Applications, 19(8–9), pp. 127–145, 1990. [Crossref]
[2] Kansa, E.J., Multiquadrics – A scattered data approximation scheme with applicationsto computational fluid dynamics II: solutions to parabolic, hyperbolic, and ellipticpartial differential equations. Computers & Mathematics with Applications, 19(8–9),pp. 147–161, 1990. [Crossref]
[3] Kansa, E.J. & Holoborodko, P., On the ill-conditioned nature of RBF strong collocation.Engineering Analysis with Boundary Elements, 78, pp. 26–30, 2017. [Crossref]
[4] http://www.exforsys.com/tutorials/c-language/decision-making-and-branching-inc.html/
[5] Chen, W., et al., Recent Advances in Radial Basis Function Collocation Methods,Springer Briefs in Applied Sciences and Technology: Berlin, 2014.
[6] Coffer, H.N. & Stadtherr, M.A., Reliability of iterative linear equation solvers in chemicalprocess simulation. Computers & Chemical Engineering, 20, pp. 1123–1132, 1996. [Crossref]
[7] Fedoseyev, A., et al., Improved multiquadric method for elliptic partial differentialequations via PDE collocation on the boundary. Computers & Mathematics with Applications,43(3–5), pp. 439–45, 2002.
[8] Wertz, J., et al., The role of the multiquadric shape parameters in solving elliptic partial differentialequations. Computers & Mathematics with Applications, 51(8), pp. 1335–1348,2006. [Crossref]
[9] Rump, S.M., Verification methods: rigorous results using floating point arithmetic. ActaNumerica, 19, pp. 287–449, 2010. [Crossref]
[10] Higham, N.J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM:Philadelphia, 2002.
[11] Brent, R. & Zimmermann, P., Modern Computer Arithmetic, Cambridge Monographson Computational and Applied Mathematics, 2010.
[12] Hida, Y.L., et al., Quad-Double Arithmetic: Algorithms, Implementation, and Application,Lawrence.Berkeley National Laboratory Technical Report LBNL-46996, 2000.
[13] Li, X.S., et al., Design, implementation and testing of extended and mixed precisionBLAS. ACM Transactions on Mathematical Software, 1, pp. 152–205, 2002. [Crossref]
[14] Higham, N.J., Multi-precision world, SIAM News, pp. 2–3, October 2017.
[15] Huang, C.S., et al., Error estimate, optimal shape factor, and high precision computationof multiquadric collocation method. Engineering Analysis with Boundary Elements,31(1), pp. 614–623, 2007. [Crossref]
[16] Huang, C.S., et al., On the increasingly flat radial basis function and optimal shapeparameter for the solution of elliptic PDEs. Engineering Analysis with Boundary Elements,34, pp. 802–809, 2010. [Crossref]
[17] Cheng, A.H.D., Multiquadric and its shape parameter - a numerical investigation oferror estimate, condition number, and round-off error by arbitrary precision computation.Engineering Analysis with Boundary Elements, 36, pp. 220–239, 2012. [Crossref]
[18] Sarra, S.A., Radial basis function approximation methods with extended precisionfloating point arithmetic. Engineering Analysis with Boundary Elements, 35, pp. 68–76,2011. [Crossref]
[19] Kansa, E.J., & Geiser, J., Numerical solution to time-dependent 4D inviscid Burgersequations. Engineering Analysis with Boundary Elements, 37, pp. 637–645, 2013. [Crossref]
[20] https://www.advanpix.com/2011/10/12/multiprecision-computationeigenvalues-eigenvector/
[21] Luh, L.T., The choice of the shape parameter – A friendly approach. EngineeringAnalysiswith Boundary Elements, 98, pp. 103–109, 2019. [Crossref]
[22] Luh, L.T., The mystery of the shape parameter III. Applied and Computational HarmonicAnalysis, 40, pp. 186–199, 2016. [Crossref]
[23] Luh, L.T., Solving Poisson’s equations by the MN curve approach. Preprint,arXiv:1905.07992 [math.NA], 2019.
[24] Madych, W.R. & Nelson, S.A., Multivariate interpolation and conditionally positivedefinite functions, II, Mathematics of Computation, 54, pp. 211–230, 1990. [Crossref]
[25] Ling, L. & Kansa, E.J., Preconditioning for radial basis functions with domain decompositionmethods. Mathematical and Computer Modelling, 40, pp. 1413–1427, 2004. [Crossref]
[26] Koczka, G., et al., An iterative domain decomposition method for solving wave propagationproblems. Electromagnetics, 34(3–4), pp. 210–221, 2014. [Crossref]
[27] Hon, Y.C. & Wu, Z., Additive Schwarz domain decomposition with radial basis approximation.International Journal of Applied Mathematics, 4, pp. 81–98, 2002.
[28] Li, J. & Hon, Y.C., Domain decomposition for radial basis meshless methods. NumericalMethods for PDEs, 20, pp. 450–462, 2004. [Crossref]
[29] Ingber, M.A., et al., A mesh free approach using radial basis functions and paralleldomain decomposition for solving three dimensional diffusion equations. InternationalJournal for Numerical Methods in Engineering, 60, pp. 2183–2201, 2004. [Crossref]
[30] Duan, Y., et al., Coupling projection domain decomposition method and Kansa’s methodin electrostatic problems. Computer Physics Communications, 180, pp. 200–214, 2009. [Crossref]
[31] Herrera, I. & Rosas Medina, A., The derived-vector space framework and four generalpurposes massively parallel DDM algorithms. Engineering Analysis with Boundary Elements,137(3), pp. 646–657, 2013. [Crossref]
[32] Hernandez-Rosales, A. & Power, H., Non-overlapping domain decomposition algorithmfor the Hermite radial basis function meshless collocation approach: applicationsto convection diffusion problems. Journal of Algorithms & Computional Technology, 1,pp. 127–159, 2007. [Crossref]
[33] Kansa, E.J. & Hon, Y.C., Circumventing the ill-conditioning problem with multiquadricradial basis functions: Applications to elliptic partial differential equations. Computers& Mathematics with Applications, 39, pp. 123–137, 2000.
[34] https://en.wikipedia.org/wiki/Schur complement method.
[35] Fornberg, B. & Flyer, N., A Primer on Radial Basis Functions with Applications to theGeosciences, SIAM: Philadelphia, PA, 2015.
[36] Gonzalez-Rodriguez, P., et al., Laurent series based RBF-FD method to avoid ill conditioning.Engineering Analysis with Boundary Elements, 52, pp. 24–31, 2015. [Crossref]
[37] Ilati, M. & Dehghan, M., The use of Radial Basis Functions (RBFs) collocation andRBF-QR methods for solving the coupled nonlinear sine-Gordon equations. EngineeringAnalysis with Boundary Elements, 52, pp. 99–109, 2015. [Crossref]
[38] Wright, G.B. & Fornberg, B., Stable computations with flat radial basis functionsusing vector-valued rational approximations. Journal of Computational Physics, 331,pp. 137–156, 2017. [Crossref]
[39] Rashidinia, J., et al., A stable method for the evaluation of Gaussian radial basis functionsolutions of interpolation and collocation problems. Computers & Mathematicswith Applications, 72, pp. 178–193, 2016. [Crossref]
[40] Fasshauer, G.E. & Zhang, J.G., On choosing “optimal” shape parameters for RBFapproximation. Numerical Algorithms, 45, pp. 345–368, 2007. [Crossref]
[41] Davis, T.A., Direct Methods for Sparse Linear Systems, SIAM: Philadelphia, 2006.
[42] Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM: Philadelphia,2003.
[43] Sarra, S.A. & Kansa, E.J., Multiquadric Radial Basis Function Approximation Methodsfor the Numerical Solution of Partial Differential Equations, Advances in ComputationalMechanics, 2, ISSN:1940-5820, 2009.
Search

Acadlore takes over the publication of IJCMEM from 2025 Vol. 13, No. 3. The preceding volumes were published under a CC BY 4.0 license by the previous owner, and displayed here as agreed between Acadlore and the previous owner. ✯ : This issue/volume is not published by Acadlore.

Open Access
Research article

Fully and Sparsely Supported Radial Basis Functions

E. J. Kansa1,
P. Holoborodko2
1
Convergent Solutions, Livermore, USA
2
Advanpix LLC, Yokohama, Japan
International Journal of Computational Methods and Experimental Measurements
|
Volume 8, Issue 3, 2020
|
Pages 208-219
Received: N/A,
Revised: N/A,
Accepted: N/A,
Available online: N/A
View Full Article|Download PDF

Abstract:

The central idea of this paper is that computer mathematics is not identical to ideal mathematics because computer numbers only have finite precision. All functions, especially the positive definite transcendental functions, are truncated and the expansion coefficients have finite precision and all branching operations require time to complete. Of all the known methods used to obtain numerical solutions to integral and partial differential equations, the global continuously differential radial basis functions (RBFs) that are implemented on computers closely resemble many aspects of ideal mathematics. The global RBFs have the attributes required to obtain very accurate numerical results for a variety of partial differential and integral equations with smooth solutions. Without the need for extremely fine discretization, the global RBFs have their spline properties and exponential convergence rates. The resulting system of full equations can be executed very rapidly on graphical processing units and field-programmable gate arrays because, with full systems, there is no branching and full systems solvers are very highly vectorized, optimizing the usage of very fast processors.

Keywords: branching, compactly supported radial basis functions, continuously differentiable, exponentially convergent, full equations, globally supported radial basis functions, multivariate, polynomials, sparse equations, splines, transcendental functions, wavelets
References
[1] Kansa, E.J., Multiquadrics - a scattered data approximation scheme with applications tocomputational fluid dynamics I: surface approximations and partial derivative estimates.Computers & Mathematics with Applications, 19(8–9), pp. 127–145, 1990. [Crossref]
[2] Kansa, E.J., Multiquadrics – A scattered data approximation scheme with applicationsto computational fluid dynamics II: solutions to parabolic, hyperbolic, and ellipticpartial differential equations. Computers & Mathematics with Applications, 19(8–9),pp. 147–161, 1990. [Crossref]
[3] Kansa, E.J. & Holoborodko, P., On the ill-conditioned nature of RBF strong collocation.Engineering Analysis with Boundary Elements, 78, pp. 26–30, 2017. [Crossref]
[4] http://www.exforsys.com/tutorials/c-language/decision-making-and-branching-inc.html/
[5] Chen, W., et al., Recent Advances in Radial Basis Function Collocation Methods,Springer Briefs in Applied Sciences and Technology: Berlin, 2014.
[6] Coffer, H.N. & Stadtherr, M.A., Reliability of iterative linear equation solvers in chemicalprocess simulation. Computers & Chemical Engineering, 20, pp. 1123–1132, 1996. [Crossref]
[7] Fedoseyev, A., et al., Improved multiquadric method for elliptic partial differentialequations via PDE collocation on the boundary. Computers & Mathematics with Applications,43(3–5), pp. 439–45, 2002.
[8] Wertz, J., et al., The role of the multiquadric shape parameters in solving elliptic partial differentialequations. Computers & Mathematics with Applications, 51(8), pp. 1335–1348,2006. [Crossref]
[9] Rump, S.M., Verification methods: rigorous results using floating point arithmetic. ActaNumerica, 19, pp. 287–449, 2010. [Crossref]
[10] Higham, N.J., Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM:Philadelphia, 2002.
[11] Brent, R. & Zimmermann, P., Modern Computer Arithmetic, Cambridge Monographson Computational and Applied Mathematics, 2010.
[12] Hida, Y.L., et al., Quad-Double Arithmetic: Algorithms, Implementation, and Application,Lawrence.Berkeley National Laboratory Technical Report LBNL-46996, 2000.
[13] Li, X.S., et al., Design, implementation and testing of extended and mixed precisionBLAS. ACM Transactions on Mathematical Software, 1, pp. 152–205, 2002. [Crossref]
[14] Higham, N.J., Multi-precision world, SIAM News, pp. 2–3, October 2017.
[15] Huang, C.S., et al., Error estimate, optimal shape factor, and high precision computationof multiquadric collocation method. Engineering Analysis with Boundary Elements,31(1), pp. 614–623, 2007. [Crossref]
[16] Huang, C.S., et al., On the increasingly flat radial basis function and optimal shapeparameter for the solution of elliptic PDEs. Engineering Analysis with Boundary Elements,34, pp. 802–809, 2010. [Crossref]
[17] Cheng, A.H.D., Multiquadric and its shape parameter - a numerical investigation oferror estimate, condition number, and round-off error by arbitrary precision computation.Engineering Analysis with Boundary Elements, 36, pp. 220–239, 2012. [Crossref]
[18] Sarra, S.A., Radial basis function approximation methods with extended precisionfloating point arithmetic. Engineering Analysis with Boundary Elements, 35, pp. 68–76,2011. [Crossref]
[19] Kansa, E.J., & Geiser, J., Numerical solution to time-dependent 4D inviscid Burgersequations. Engineering Analysis with Boundary Elements, 37, pp. 637–645, 2013. [Crossref]
[20] https://www.advanpix.com/2011/10/12/multiprecision-computationeigenvalues-eigenvector/
[21] Luh, L.T., The choice of the shape parameter – A friendly approach. EngineeringAnalysiswith Boundary Elements, 98, pp. 103–109, 2019. [Crossref]
[22] Luh, L.T., The mystery of the shape parameter III. Applied and Computational HarmonicAnalysis, 40, pp. 186–199, 2016. [Crossref]
[23] Luh, L.T., Solving Poisson’s equations by the MN curve approach. Preprint,arXiv:1905.07992 [math.NA], 2019.
[24] Madych, W.R. & Nelson, S.A., Multivariate interpolation and conditionally positivedefinite functions, II, Mathematics of Computation, 54, pp. 211–230, 1990. [Crossref]
[25] Ling, L. & Kansa, E.J., Preconditioning for radial basis functions with domain decompositionmethods. Mathematical and Computer Modelling, 40, pp. 1413–1427, 2004. [Crossref]
[26] Koczka, G., et al., An iterative domain decomposition method for solving wave propagationproblems. Electromagnetics, 34(3–4), pp. 210–221, 2014. [Crossref]
[27] Hon, Y.C. & Wu, Z., Additive Schwarz domain decomposition with radial basis approximation.International Journal of Applied Mathematics, 4, pp. 81–98, 2002.
[28] Li, J. & Hon, Y.C., Domain decomposition for radial basis meshless methods. NumericalMethods for PDEs, 20, pp. 450–462, 2004. [Crossref]
[29] Ingber, M.A., et al., A mesh free approach using radial basis functions and paralleldomain decomposition for solving three dimensional diffusion equations. InternationalJournal for Numerical Methods in Engineering, 60, pp. 2183–2201, 2004. [Crossref]
[30] Duan, Y., et al., Coupling projection domain decomposition method and Kansa’s methodin electrostatic problems. Computer Physics Communications, 180, pp. 200–214, 2009. [Crossref]
[31] Herrera, I. & Rosas Medina, A., The derived-vector space framework and four generalpurposes massively parallel DDM algorithms. Engineering Analysis with Boundary Elements,137(3), pp. 646–657, 2013. [Crossref]
[32] Hernandez-Rosales, A. & Power, H., Non-overlapping domain decomposition algorithmfor the Hermite radial basis function meshless collocation approach: applicationsto convection diffusion problems. Journal of Algorithms & Computional Technology, 1,pp. 127–159, 2007. [Crossref]
[33] Kansa, E.J. & Hon, Y.C., Circumventing the ill-conditioning problem with multiquadricradial basis functions: Applications to elliptic partial differential equations. Computers& Mathematics with Applications, 39, pp. 123–137, 2000.
[34] https://en.wikipedia.org/wiki/Schur complement method.
[35] Fornberg, B. & Flyer, N., A Primer on Radial Basis Functions with Applications to theGeosciences, SIAM: Philadelphia, PA, 2015.
[36] Gonzalez-Rodriguez, P., et al., Laurent series based RBF-FD method to avoid ill conditioning.Engineering Analysis with Boundary Elements, 52, pp. 24–31, 2015. [Crossref]
[37] Ilati, M. & Dehghan, M., The use of Radial Basis Functions (RBFs) collocation andRBF-QR methods for solving the coupled nonlinear sine-Gordon equations. EngineeringAnalysis with Boundary Elements, 52, pp. 99–109, 2015. [Crossref]
[38] Wright, G.B. & Fornberg, B., Stable computations with flat radial basis functionsusing vector-valued rational approximations. Journal of Computational Physics, 331,pp. 137–156, 2017. [Crossref]
[39] Rashidinia, J., et al., A stable method for the evaluation of Gaussian radial basis functionsolutions of interpolation and collocation problems. Computers & Mathematicswith Applications, 72, pp. 178–193, 2016. [Crossref]
[40] Fasshauer, G.E. & Zhang, J.G., On choosing “optimal” shape parameters for RBFapproximation. Numerical Algorithms, 45, pp. 345–368, 2007. [Crossref]
[41] Davis, T.A., Direct Methods for Sparse Linear Systems, SIAM: Philadelphia, 2006.
[42] Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM: Philadelphia,2003.
[43] Sarra, S.A. & Kansa, E.J., Multiquadric Radial Basis Function Approximation Methodsfor the Numerical Solution of Partial Differential Equations, Advances in ComputationalMechanics, 2, ISSN:1940-5820, 2009.
Cite this:
APA Style
IEEE Style
BibTex Style
MLA Style
Chicago Style
GB-T-7714-2015
Kansa, E. J. & Holoborodko, P. (2020). Fully and Sparsely Supported Radial Basis Functions. Int. J. Comput. Methods Exp. Meas., 8(3), 208-219. https://doi.org/10.2495/CMEM-V8-N3-208-219
E. J. Kansa and P. Holoborodko, "Fully and Sparsely Supported Radial Basis Functions," Int. J. Comput. Methods Exp. Meas., vol. 8, no. 3, pp. 208-219, 2020. https://doi.org/10.2495/CMEM-V8-N3-208-219
@research-article{Kansa2020FullyAS,
title={Fully and Sparsely Supported Radial Basis Functions},
author={E. J. Kansa and P. Holoborodko},
journal={International Journal of Computational Methods and Experimental Measurements},
year={2020},
page={208-219},
doi={https://doi.org/10.2495/CMEM-V8-N3-208-219}
}
E. J. Kansa, et al. "Fully and Sparsely Supported Radial Basis Functions." International Journal of Computational Methods and Experimental Measurements, v 8, pp 208-219. doi: https://doi.org/10.2495/CMEM-V8-N3-208-219
E. J. Kansa and P. Holoborodko. "Fully and Sparsely Supported Radial Basis Functions." International Journal of Computational Methods and Experimental Measurements, 8, (2020): 208-219. doi: https://doi.org/10.2495/CMEM-V8-N3-208-219
KANSA E J， HOLOBORODKO P. Fully and Sparsely Supported Radial Basis Functions[J]. International Journal of Computational Methods and Experimental Measurements， 2020， 8(3)： 208-219. https://doi.org/10.2495/CMEM-V8-N3-208-219