Skip to main content
SpringerLink
Log in
Book cover

Recent Advances in Radial Basis Function Collocation Methods pp 5–28Cite as

Radial Basis Functions

  • Chapter
  • First Online:

Part of the book series: SpringerBriefs in Applied Sciences and Technology ((BRIEFSAPPLSCIENCES))

Abstract

The traditional basis functions in numerical PDEs are mostly coordinate functions, such as polynomial and trigonometric functions, which are computationally expensive in dealing with high dimensional problems due to their dependency on geometric complexity. Alternatively, radial basis functions (RBFs) are constructed in terms of one-dimensional distance variable irrespective of dimensionality of problems and appear to have a clear edge over the traditional basis functions directly in terms of coordinates. In the first part of this chapter, we introduces classical RBFs, such as globally-supported RBFs (Polyharmonic splines, Multiquadratics, Gaussian, etc.), and recently developed RBFs, such as compactly-supported RBFs. Following this, several problem-dependent RBFs, such as fundamental solutions, general solutions, harmonic functions, and particular solutions, are presented. Based on the second Green identity, we propose the kernel RBF-creating strategy to construct the appropriate RBFs.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. E. Folio, Distance Transform. In: Technical Report no 0806, revision 1748. Laboratoire de Recherche et Dveloppement de l’Epita (Le Kremlin-Bictre cedex-France, 2008)

    Google Scholar 

  2. M.D. Buhmann, Radial Basis Function: Theory and Implementations (Cambridge University Press, Cambridge, 2003)

    Google Scholar 

  3. R. Franke, Scattered data interpolation: tests of some method. Math. Comput. 38(157), 181–200 (1982)

    MathSciNet  MATH  Google Scholar 

  4. J.G. Wang, G.R. Liu, On the optimal shape parameters of radial basis function used for 2-D meshless methods. Comput. Methods Appl. Mech. Eng. 191(23–24), 2611–2630 (2002)

    Article  MATH  Google Scholar 

  5. A.H.D. Cheng, M.A. Golberg, E.J. Kansa, G. Zammito, Exponential convergence and H-c multiquadric collocation method for partial differential equations. Numer. Methods Part. Differ. Eq. 19(5), 571–594 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. C.S. Huang, C.F. Lee, A.H.D. Cheng, Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Eng. Anal. Boundary Elem. 31(7), 614–623 (2007)

    Article  MATH  Google Scholar 

  7. C.M.C. Roque, A.J.M. Ferreira, Numerical experiments on optimal shape parameters for radial basis function. Numer. Methods Part. Differ. Eq. 26(3), 675–689 (2010)

    MathSciNet  MATH  Google Scholar 

  8. B. Fornberg, C. Piret, On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere. J. Comput. Phys. 227(5), 2758–2780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Revue Francaise D Automatique Informatique Recherche Operationnelle 10(12), 5–12 (1976)

    MathSciNet  Google Scholar 

  10. W.R. Madych, S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. Approx. Theor. Appl. 4, 77–89 (1988)

    MathSciNet  MATH  Google Scholar 

  11. W.R. Madych, S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. 2. Math. Comput. 54(189), 211–230 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. W.R. Madych, S.A. Nelson, Bounds on multivariate polynomials and exponential error-estimates for multiquadric interpolation. J. Approx. Theor. 70(1), 94–114 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Z.M. Wu, R.S. Schaback, Local error-estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13(1), 13–27 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. A.H.D. Cheng, Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation. Eng. Anal. Boundary Elem. 36, 220–239 (2012)

    Article  MATH  Google Scholar 

  15. H. Wendland, Scattered Data Approximation (Cambridge University Press, Cambridge, 2005)

    Google Scholar 

  16. W.R. Madych, Miscellaneous error-bounds for multiquadric and related interpolators. Comput. Math. Appl. 24(12), 121–138 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. W.K. Liu, J. Sukky, Multiple-scale reproducing kernel particle methods for large deformation problems. Int. J. Numer. Meth. Eng. 41(7), 1339–1362 (1998)

    Article  MATH  Google Scholar 

  18. Z. Wu, Compactly supported positive definite radial functions. Adv. Comput. Math. 4(1), 283–292 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4(1), 389–396 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. M.D. Buhmann, Radial functions on compact support, in Proceedings of the Edinburgh Mathematical Society (Series 2) 41(01), 33–46 (1998)

    Google Scholar 

  21. K.K. Prem, Fundamental Solutions for Differential Operators and Applications. (Birkhauser Boston Inc., Cambridge, 1996)

    Google Scholar 

  22. A.J. Nowak, A.C. Neves, The Multiple Reciprocity Boundary Element Method. (Computational Mechanics Publication, Southampton, 1994)

    Google Scholar 

  23. M. Itagaki, Higher order three-dimensional fundamental solutions to the Helmholtz and the modified Helmholtz equations. Eng. Anal. Boundary Elem. 15, 289–293 (1995)

    Article  Google Scholar 

  24. W. Chen, Z.J. Shen, L.J. Shen, G.W. Yuan, General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates. Eng. Anal. Boundary Elem. 29(7), 699–702 (2005)

    Article  MATH  Google Scholar 

  25. W. Chen, Meshfree boundary particle method applied to Helmholtz problems. Eng. Anal. Boundary Elem. 26(7), 577–581 (2002)

    Article  MATH  Google Scholar 

  26. W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for Poisson equations. Eng. Anal. Boundary Elem. 29(8), 756–760 (2005)

    Article  MATH  Google Scholar 

  27. Y.C. Hon, Z. Wu, A numerical computation for inverse boundary determination problem. Eng. Anal. Boundary Elem. 24(7–8), 599–606 (2000)

    Article  MATH  Google Scholar 

  28. E.J. Kansa, Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. Z.J. Fu, W. Chen, A truly boundary-only Meshfree method applied to Kirchhoff plate bending problems. Adv. Appl. Math. Mech. 1(3), 341–352 (2009)

    MathSciNet  Google Scholar 

  30. W. Chen, Z.J. Fu, B.T. Jin, A truly boundary-only Meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique. Eng. Anal. Boundary Elem. 34(3), 196–205 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. K.E. Atkinson, The numerical evaluation of particular solutions for Poisson’s equation. IMA J. Numer. Anal. 5, 319–338 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. C.S. Chen, Y.F. Rashed, Evaluation of thin plate spline based particular solutions for helmholtz-type operators for the drm. Mech. Res. Commun. 25, 195–201 (1998)

    Article  MATH  Google Scholar 

  33. A.S. Muleshkov, M.A. Golberg, C.S. Chen, Particular solutions of helmholtz-type operators using higher order polyharmonic splines. Comput. Mech. 24, 411–419 (1999)

    Article  MathSciNet  Google Scholar 

  34. A.H.D. Cheng, Particular solutions of Laplacian, helmholtz-type, and polyharmonic operators involving higher order radial basis function. Eng. Anal. Boundary Elem. 24, 531–538 (2000)

    Article  MATH  Google Scholar 

  35. A.S. Muleshkov, M.A. Golberg, Particular solutions of the multi-helmholtz-type equation. Eng. Anal. Boundary Elem. 31, 624–630 (2007)

    Article  MATH  Google Scholar 

  36. C.S. Chen, Y.C. Hon, R.S. Schaback, Radial basis function with Scientific Computation (University of Southern Mississippi, Mississippi, 2007)

    Google Scholar 

  37. C.C. Tsai, Particular solutions of splines and monomials for polyharmonic and products of Helmholtz operators. Eng. Anal. Boundary Elem. 33(4), 514–521 (2009)

    Article  MATH  Google Scholar 

  38. G.M. Yao, Local Radial Basis Function Methods for Solving Partial Differential Equations. Ph.D. Dissertation, University of Southern Mississippi, 2010

    Google Scholar 

  39. R.E. Carlson, T.A. Foley, Interpolation of track data with radial basis methods. Comput. Math. Appl. 24, 27–34 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Y.C. Shiah, C.L. Tan, BEM treatment of three-dimensional anisotropic field problems by direct domain mapping. Eng. Anal. Boundary Elem. 28(1), 43–52 (2004)

    Article  MATH  Google Scholar 

  41. B.T. Jin, W. Chen, Boundary knot method based on geodesic distance for anisotropic problems. J. Comput. Phys. 215(2), 614–629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. D.E. Myers, S. De Iaco, D. Posa, L. De Cesare, Space-time radial basis function. Comput. Math. Appl. 43(3–5), 539–549 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. W. Chen, New RBF collocation schemes and kernel RBF with applications. Lect. Notes Comput. Sci. Eng. 26, 75–86 (2002)

    Article  Google Scholar 

  44. D.L. Young, C.C. Tsai, K. Murugesan, C.M. Fan, C.W. Chen, Time-dependent fundamental solutions for homogeneous diffusion problems. Eng. Anal. Boundary Elem. 28(12), 1463–1473 (2004)

    Article  MATH  Google Scholar 

  45. C.C. Tsai, D.L. Young, C.M. Fan, C.W. Chen, MFS with time-dependent fundamental solutions for unsteady Stokes equations. Eng. Anal. Boundary Elem. 30(10), 897–908 (2006)

    Article  MATH  Google Scholar 

  46. W. Chen, M. Tanaka, New Insights into Boundary-only and Domain-type RBF Methods. Int. J. Nonlinear Sci. Numer. Simul. 1(3), 145–151 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. W. Chen, M. Tanaka, Relationship between boundary integral equation and radial basis function. Paper presented at the 52th symposium of Japan society for computational methods in engineering (JASCOME) on BEM, Tokyo

    Google Scholar 

  48. W. Chen, M. Tanaka, A meshless, integration-free, and boundary-only RBF technique. Comput. Math. Appl. 43(3–5), 379–391 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  49. J. Lin, W. Chen, K.Y. Sze, A new radial basis function for Helmholtz problems. Eng. Anal. Boundary Elem. 36(12), 1923–1930 (2012)

    Article  MathSciNet  Google Scholar 

  50. S.G. Ahmed, A collocation method using new combined radial basis function of thin plate and multiquadraic types. Eng. Anal. Boundary Elem. 30(8), 697–701 (2006)

    Article  MATH  Google Scholar 

  51. B. Fornberg, E. Larsson, G. Wright, A new class of oscillatory radial basis function. Comput. Math. Appl. 51(8), 1209–1222 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  52. V. Kompis, M. Stiavnicky, M. Zmindak, Z. Murcinkova, Trefftz radial basis function (TRBF), in Proceedings of Lsame.08: Leuven Symposium on Applied Mechanics in Engineering, Pts 1 and 2, 25–35 (2008)

    Google Scholar 

  53. N.A. Libre, A. Emdadi, E.J. Kansa, M. Shekarchi, M. Rahimian, Wavelet based adaptive RBF method for nearly singular poisson-type problems on irregular domains. CMES Comput. Model. Eng. Sci. 50(2), 161–190 (2009)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mechanics and Materials, Hohai University, Nanjing, People’s Republic of China

    Wen Chen

  2. College of Mechanics and Materials, Hohai University, Nanjing, People’s Republic of China

    Zhuo-Jia Fu

  3. University of Southern Mississippi, Hattiesburg, MS, USA

    C. S. Chen

Authors
  1. Wen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhuo-Jia Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. S. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen Chen .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Author(s)

About this chapter

Cite this chapter

Chen, W., Fu, ZJ., Chen, C.S. (2014). Radial Basis Functions. In: Recent Advances in Radial Basis Function Collocation Methods. SpringerBriefs in Applied Sciences and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39572-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39572-7_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39571-0

  • Online ISBN: 978-3-642-39572-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation