Journal of Scientific Computing

, Volume 44, Issue 3, pp 286–300 | Cite as

Radial Basis Function Interpolation on Irregular Domain through Conformal Transplantation

Article

Abstract

In this paper, Radial Basis Function (RBF) method for interpolating two dimensional functions with localized features defined on irregular domain is presented. RBF points located inside the domain and on its boundary are chosen such that they are the image of conformally mapped points on concentric circles on a unit disk. On the disk, a fast RBF solver to compute RBF coefficients developed by Karageorghis et al. (Appl. Numer. Math. 57(3):304–319, 2007) is used. Approximation values at desired points in the domain can be computed through the process of conformal transplantation. Some numerical experiments are given in a style of a tutorial and MATLAB code that solves RBF coefficients using up to 100,000 RBF points is provided.

Keywords

Radial basis functions Schwarz-Christoffel mapping Meshfree Interpolation 

References

  1. 1.
    Amestoy, P.R., Enseeiht-Irit, Davis, T.A., Duff, I.S.: Algorithm 837: Amd, an approximate minimum degree ordering algorithm. ACM Trans. Math. Softw. 30(3), 381–388 (2004) MATHCrossRefGoogle Scholar
  2. 2.
    Banjai, L.: Eigenfrequencies of fractal drums. J. Comput. Appl. Math. 198(1), 1–18 (2007) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beatson, R.K., Cherrie, J.B., Mouat, C.T.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11(2–3), 253–270 (1999) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge Monographs on Applied and Computational Mathematics, vol. 12. Cambridge University Press, Cambridge (2003) MATHGoogle Scholar
  5. 5.
    Buhmann, M.D., Dyn, N.: Spectral convergence of multiquadric interpolation. Proc. Edinb. Math. Soc. (2) 36(2), 319–333 (1993) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Casciola, G., Montefusco, L.B., Morigi, S.: The regularizing properties of anisotropic radial basis functions. Appl. Math. Comput. 190(2), 1050–1062 (2007) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cheng, A.H.-D., Golberg, M.A., Kansa, E.J., Zammito, G.: Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numer. Methods Partial Differ. Equ. 19(5), 571–594 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Driscoll, T.A.: Algorithm 843: improvements to the Schwarz-Christoffel toolbox for MATLAB. ACM Trans. Math. Softw. 31(2), 239–251 (2005) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Driscoll, T.A., Fornberg, B.: Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43, 413–422 (2002) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Driscoll, T.A., Heryudono, A.R.H.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53, 927–939 (2007) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A Driscoll, T., Trefethen, L.N.: Schwarz-Christoffel Mapping. Cambridge Monographs on Applied and Computational Mathematics, vol. 8. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
  12. 12.
    Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Co. Pte. Ltd., Hackensack (2007). With 1 CD-ROM (Windows, Macintosh and UNIX) MATHGoogle Scholar
  13. 13.
    Faul, A.C., Goodsell, G., Powell, M.J.D.: A Krylov subspace algorithm for multiquadric interpolation in many dimensions. IMA J. Numer. Anal. 25(1), 1–24 (2005) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge Monographs on Applied and Computational Mathematics, vol. 1. Cambridge University Press, Cambridge (1996) MATHGoogle Scholar
  15. 15.
    Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions in 2-D. Uppsala University Technical Report, 2009 Google Scholar
  16. 16.
    Fornberg, B., Piret, C.: A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput. 30(1), 60–80 (2007/2008) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Fornberg, B., Wright, G.: Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48(5–6), 853–867 (2004) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Fornberg, B., Zuev, J.: The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput. Math. Appl. 54(3), 379–398 (2007) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Henrici, P.: Applied and Computational Complex Analysis. Wiley Classics Library, vol. 1. John Wiley & Sons Inc., New York (1988). Power series—integration—conformal mapping—location of zeros. Reprint of the 1974 original, A Wiley-Interscience Publication MATHGoogle Scholar
  20. 20.
    Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007) MATHCrossRefGoogle Scholar
  21. 21.
    Jung, J.-H., Durante, V.: An iteratively adaptive multiquardic radial basis function method for detection of local jump discontinuities. Appl. Numer. Math. 59, 1449–1466 (2009) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Jung, J.-H., Gottlieb, S., Kim, S.O.: Two-dimensional edge detection based on the adaptive iterative MQ-RBF method. Appl. Numer. Math. (submitted) Google Scholar
  23. 23.
    Kansa, E.J.: 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) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Karageorghis, A., Chen, C.S., Smyrlis, Y.-S.: A matrix decomposition RBF algorithm: approximation of functions and their derivatives. Appl. Numer. Math. 57(3), 304–319 (2007) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46(5–6), 891–902 (2003) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Platte, R.B.: How fast do radial basis function interpolants of analytic functions converge? IMA J. Numer. Anal. (submitted) Google Scholar
  27. 27.
    Platte, R.B., Driscoll, T.A.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43(2), 750–766 (2005) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Roussos, G., Baxter, B.J.C.: Rapid evaluation of radial basis functions. J. Comput. Appl. Math. 180(1), 51–70 (2005) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sarra, S.A.: Adaptive radial basis function methods for time dependent partial differential equations. Appl. Numer. Math. 54(1), 79–94 (2005) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Schaback, R.: Limit problems for interpolation by analytic radial basis functions. J. Comput. Appl. Math. 212, 127–149 (2008) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Trefethen, L.N.: Spectral Methods in MATLAB. Software, Environments, and Tools, vol. 10. SIAM, Philadelphia (2000) MATHGoogle Scholar
  33. 33.
    Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005) MATHGoogle Scholar
  34. 34.
    Yoon, J.: Spectral approximation orders of radial basis function interpolation on the Sobolev space. SIAM J. Math. Anal. 33(4), 946–958 (2001) (electronic) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Massachusetts DartmouthDartmouthUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

Personalised recommendations