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Solving Partial Differential Equations with Multiscale Radial Basis Functions

  • Holger WendlandEmail author
Chapter

Abstract

The goal of this paper is to review, discuss and extend the current theory on multiscale radial basis functions for solving elliptic partial differential equations. Multiscale radial basis functions provide approximation spaces using different scales and shifts of a compactly supported, positive definite function in an orderly fashion. In this paper, both collocation and Galerkin approximation are described and analysed. To this end, first symmetric and non-symmetric recovery is discussed. Then, error estimates for both schemes are derived, though special emphasis is given to Galerkin approximation, since the current situation here is not as clear as in the case of collocation. We will distinguish between stationary and non-stationary multiscale approximation spaces and multilevel approximation schemes. For Galerkin approximation, we will establish error bounds in the stationary setting based upon Cea’s lemma showing that the approximation spaces are indeed rich enough. Unfortunately, convergence of a simple residual correction algorithm, which is often applied in this context to compute the approximation, can only be shown for a non-stationary multiscale approximation space.

References

  1. 1.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (1994)CrossRefGoogle Scholar
  2. 2.
    Buhmann, M.D.: Radial basis functions. In: Acta Numerica 2000, vol. 9, pp. 1–38. Cambridge University Press, Cambridge (2000)Google Scholar
  3. 3.
    Buhmann, M.D.: Radial Basis Functions. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  4. 4.
    Chen, C.S., Ganesh, M., Golberg, M.A., Cheng, A.H.D.: Multilevel compact radial functions based computational schemes for some elliptic problems. Comput. Math. Appl. 43, 359–378 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chernih, A., Le Gia, Q.T.: Multiscale methods with compactly supported radial basis functions for elliptic partial differential equations on bounded domains. ANZIAM J. (E) 54, C137–C152 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chernih, A., Le Gia, Q.T.: Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains. Adv. Comput. Math. 42, 1187–1208 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chernih, A., Le Gia, Q.T.: Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs. IMA J. Numer. Anal. 34, 569–591 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Farrell, P., Wendland, H.: RBF multiscale collocation for second order elliptic boundary value problems. SIAM J. Numer. Anal. 51, 2403–2425 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Farrell, P., Gillow, K., Wendland, H.: Multilevel interpolation of divergence-free vector fields. IMA J. Numer. Anal. 37, 332–353 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fasshauer, G.E.: Solving partial differential equations by collocation with radial basis functions. In: Méhauté, A.L., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 131–138. Vanderbilt University Press, Nashville (1997)Google Scholar
  11. 11.
    Fasshauer, G.E.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific, Singapore (2007)CrossRefGoogle Scholar
  13. 13.
    Fasshauer, G.E., Jerome, J.W.: Multistep approximation algorithms: improved convergence rates through postconditioning with smoothing kernels. Adv. Comput. Math. 10, 1–27 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferrari, S., Maggioni, M., Borhese, N.A.: Multiscale approximation with hierarchical radial basis functions networks. IEEE Trans. Neural Netw. 15, 178–188 (2004)CrossRefGoogle Scholar
  15. 15.
    Floater, M.S., Iske, A.: Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math. 73, 65–78 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. In: Iserles, A. (ed.) Acta Numerica, vol. 24, pp. 215–258. Cambridge University Press, Cambridge (2015)Google Scholar
  17. 17.
    Franke, C., Schaback, R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 381–399 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Franke, C., Schaback, R.: Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput. 93, 73–82 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Giesl, P., Wendland, H.: Meshless collocation: Error estimates with application to dynamical systems. SIAM J. Numer. Anal. 45, 1723–1741 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hales, S.J., Levesley, J.: Error estimates for multilevel approximation using polyharmonic splines. Numer. Algoritm. 30, 1–10 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Heuer, N., Tran, T.: A mixed method for Dirichlet problems with radial basis functions. Comput. Math. Appl. 66, 2045–2055 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hon, Y.C., Schaback, R.: On unsymmetric collocation by radial basis functions. J. Appl. Math. Comput. 119, 177–186 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Le Gia, Q.T., Wendland, H.: Data compression on the sphere using multiscale radial basis functions. Adv. Comput. Math. 40, 923–943 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Le Gia, Q.T., Sloan, I., Wendland, H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal. 48, 2065–2090 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Le Gia, Q.T., Sloan, I., Wendland, H.: Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere. Appl. Comput. Harmon. Anal. 32, 401–412 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Le Gia, Q.T., Sloan, I., Wendland, H.: Multiscale RBF collocation for solving PDEs on spheres. Numer. Math. 121, 99–125 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Le Gia, Q.T., Sloan, I.H., Wendland, H.: Zooming from global to local: a multiscale RBF approach. Adv. Comput. Math. 43, 581–606 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li, M., Cao, F.: Multiscale interpolation on the sphere: convergence rate and inverse theorem. Appl. Math. Comput. 263, 134–150 (2015)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Morton, T.M., Neamtu, M.: Error bounds for solving pseudodifferential equatons on spheres by collocation with zonal kernels. J. Approx. Theory 114, 242–268 (2002)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Narcowich, F.J., Schaback, R., Ward, J.D.: Multilevel interpolation and approximation. Appl. Comput. Harmon. Anal. 7, 243–261 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24, 175–186 (2006)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ohtake, Y., Belyaev, A., Alexa, M., Turk, G., Seidel, H.P.: Multi-level partition of unity implicits. ACM Trans. Graph. 22, 463–470 (2003)CrossRefGoogle Scholar
  33. 33.
    Ron, A.: The L 2-approximation orders of principal shift-invariant spaces generated by a radial basis function. In: Braess, D., et al. (eds.) Numerical Methods in Approximation Theory. vol. 9: Proceedings of the Conference Held in Oberwolfach, Germany, 24–30 Nov 1991. International Series of Numerican Mathematics, vol. 105, pp. 245–268. Birkhäuser, Basel (1992)CrossRefGoogle Scholar
  34. 34.
    Schaback, R.: Creating surfaces from scattered data using radial basis functions. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces, pp. 477–496. Vanderbilt University Press, Nashville (1995)zbMATHGoogle Scholar
  35. 35.
    Schaback, R., Wendland, H.: Kernel techniques: from machine learning to meshless methods. In: Iserles, A. (ed.) Acta Numerica, vol. 15, pp. 543–639. Cambridge University Press, Cambridge (2006)Google Scholar
  36. 36.
    Townsend, A., Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains with zero boundary values. IMA J. Numer. Anal. 33, 1095–1114 (2013)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wendland, H.: Numerical solutions of variational problems by radial basis functions. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX, vol. 2: Computational Aspects, pp. 361–368. Vanderbilt University Press, Nashville (1998)Google Scholar
  39. 39.
    Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  40. 40.
    Wendland, H.: On the stability of meshless symmetric collocation for boundary value problems. BIT 47, 455–468 (2007)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains. Numer. Math. 116, 493–517 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wendland, H.: Multiscale radial basis functions. In: Pesenson, I., Gia, Q.T.L., Mayeli, A., Mhaskar, H., Zhou, D.X. (eds.) Frames and Other Bases in Abstract and Function Spaces – Novel Methods in Harmonic Analysis, vol. 1, pp. 265–299. Birkhäuser, Cham (2017)Google Scholar
  43. 43.
    Wu, Z.: Compactly supported positive definite radial functions. Adv. Comput. Math. 4, 283–292 (1995)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Xu, B., Lu, S., Zhong, M.: Multiscale support vector regression method in Sobolev spaces on bounded domains. Appl. Anal. 94, 548–569 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BayreuthBayreuthGermany

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