RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

  • S. De Marchi
  • A. Martínez
  • E. PerracchioneEmail author
  • M. Rossini


We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure.


Partition of unity method Radial basis functions Meshfree approximation Elliptic PDEs Variably scaled kernels 

Mathematics Subject Classification

65D05 65D15 65N99 



We sincerely thank the reviewers for their insightful comments. This research has been accomplished within Rete ITaliana di Approssimazione (RITA) and supported by GNCS-IN\(\delta \)AM. The first author was partially supported by the research project Approximation by radial basis functions and polynomials: applications to CT, MPI and PDEs on manifolds, No. DOR1695473. The third author was partially supported by the research project Radial basis functions approximations: stability issues and applications, No. BIRD167404.


  1. 1.
    Bozzini, M., Lenarduzzi, L., Rossini, M.: Polyharmonic splines: an approximation method for noisy scattered data of extra-large size. Appl. Math. Comput. 216, 317–331 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199–219 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caliari, M., De Marchi, S., Vianello, M.: Bivariate polynomial interpolation on the square at new nodal sets. Appl. Math. Comput. 165, 261–274 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cancelliere, R., Gai, M., Gallinari, P., Rubini, L.: OCReP: an optimally conditioned regularization for pseudoinversion based neural training. Neural Netw. 71, 76–87 (2015)CrossRefGoogle Scholar
  5. 5.
    Cavoretto, R., De Rossi, A., Dell’Accio, F., Di Tommaso, F.: Fast computation of triangular Shepard interpolants. J. Comput. Appl. Math. (2018).
  6. 6.
    Cavoretto, R., De Rossi, A., Perracchione, E.: Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71, 2568–2584 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cavoretto, R., De Rossi, A., Perracchione, E., Venturino, E.: Graphical representation of separatrices of attraction basins in two and three-dimensional dynamical systems. Int. J. Comput. Methods 14, 1750008 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cavoretto, R., Fasshauer, G.E., McCourt, M.: An introduction to the Hilbert–Schmidt SVD using iterated Brownian bridge kernels. Numer. Algorithms 68, 393–422 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Davydov, O., Oanh, D.T.: Adaptive meshless centres and RBF stencils for Poisson equation. J. Comput. Phys. 304, 230–287 (2011)MathSciNetzbMATHGoogle Scholar
  10. 10.
    De Marchi, S.: On optimal center locations for radial basis interpolation: computational aspects. Rend. Sem. Mat. Torino 61, 343–358 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    De Marchi, S., Idda, A., Santin, G.: A rescaled method for RBF approximation. In: Fasshauer, G.E., et al. (eds.) Approximation Theory XV: San Antonio 2016, vol. 201, pp. 39–59. Springer, New York (2017)CrossRefGoogle Scholar
  12. 12.
    De Marchi, S., Martínez, A., Perracchione, E.: Fast and stable rational RBF-based partition of unity interpolation. J. Comput. App. Math. (2018).
  13. 13.
    De Marchi, S., Santin, G.: Fast computation of orthonormal basis for RBF spaces through Krylov space methods. BIT 55, 949–966 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De Rossi, A., Perracchione, E., Venturino, E.: Fast strategy for PU interpolation: an application for the reconstruction of separatrix manifolds. Dolom. Res. Notes Approx. 9, 3–12 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fasshauer, G.E.: Dealing with Ill-conditioned RBF systems. Dolomites Res. Notes Approx. 1 (2008).
  17. 17.
    Fasshauer, G.E.: Meshfree Approximations Methods with Matlab. World Scientific, Singapore (2007)CrossRefGoogle Scholar
  18. 18.
    Fasshauer, G.E., Zhang, J.G.: On choosing “optimal” shape parameters for RBF approximation. Numer. Algorithms 45, 345–368 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Farrell, P., Wendland, H.: RBF multiscale collocation for second order elliptic boundary value problems. J. Numer. Anal. 51, 2403–2425 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33, 869–892 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Francomano, E., Hilker, F.M., Paliaga, M., Venturino, E.: An efficient method to reconstruct invariant manifolds of saddle points. Dolom. Res. Notes Approx. 10, 25–30 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fuhry, M., Reichel, L.: A new Tikhonov regularization method. Numer. Algorithms 59, 433–445 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Heryudono, A., Larsson, E., Ramage, A., Von Sydow, L.: Preconditioning for radial basis function partition of unity methods. J. Sci. Comput. 67, 1089–1109 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hon, Y.C., Schaback, R.: On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119, 177–186 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hon, Y.C., Schaback, R., Zhou, X.: An adaptive greedy algorithm for solving large RBF collocation problems. Numer. Algorithms 32, 13–25 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kansa, E.J.: Application of Hardy’s multiquadric interpolation to hydrodynamics. In: Proceedings of 1986 Simulation Conference, vol. l4, pp. 111–117 (1986)Google Scholar
  27. 27.
    Kowalewski, M., Larsson, E., Heryudono, A.: An adaptive interpolation scheme for molecular potential energy surfaces. J. Chem. Phys. 145, 84–104 (2016)CrossRefGoogle Scholar
  28. 28.
    Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891–902 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Larsson, E., Lehto, E., Heryudono, A., Fornberg, B.: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J. Sci. Comput. 35, A2096–A2119 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Larsson, E., Shcherbakov, V., Heryudono, A.: A least squares radial basis function partition of unity method for solving PDEs. SIAM J. Sci. Comput. 39, A2538–A2563 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ling, L., Kansa, E.J.: A least-squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23, 31–54 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ling, L., Opfer, R., Schaback, R.: Results on meshless collocation techniques. Eng. Anal. Bound. Elem. 30, 247–253 (2006)CrossRefGoogle Scholar
  33. 33.
    Melenk, J.M., Babu\(\check{\text{s}}\)ka, I., Basic theory and applications: The partition of unity finite element method. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996)Google Scholar
  34. 34.
    Oanh, D.T., Davydov, O., Phu, H.X: Adaptive RBF-FD method for elliptic problems with point singularities in 2D. Preprint (2016)Google Scholar
  35. 35.
    Pazouki, M., Schaback, R.: Bases for kernel-based spaces. J. Comput. Appl. Math. 236, 575–588 (2011)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Romani, L., Rossini, M., Schenone, D.: Edge detection methods based on RBF interpolation. J. Comput. Appl. Math. (2018).
  37. 37.
    Rossini, M.: Interpolating functions with gradient discontinuities via variably scaled kernels. Dolom. Res. Notes Approx. 11, 3–14 (2018)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. J. Sci. Comput. 64, 341–367 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Santin, G., Haasdonk, B.: Convergence rate of the data-independent \(P\)-greedy algorithm in kernel-based approximation. In: Dolomites Research Notes on Approximation, vol. 10, pp. 68–78. Special issue (2017)Google Scholar
  40. 40.
    Sarra, S.A.: The Matlab radial basis function toolbox. J. Open Res. Softw. 5, 1–10 (2017)CrossRefGoogle Scholar
  41. 41.
    Sarra, S.A., Bay, Y.: A rational radial basis function method for accurately resolving discontinuities and steep gradients. Preprint (2017)Google Scholar
  42. 42.
    Shepard, D.: A two-dimensional interpolation function for irregularly spaced data. In: Proceedings of 23-rd National Conference, Brandon/Systems Press, Princeton, pp. 517–524 (1968)Google Scholar
  43. 43.
    Schaback, R.: Convergence of unsymmetric kernel-based meshless collocation methods. SIAM J. Numer. Anal. 45, 333–351 (2007)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Shcherbakov, V., Larsson, E.: Radial basis function partition of unity methods for pricing vanilla basket options. Comput. Math. Appl. 71, 185–200 (2016)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963)zbMATHGoogle Scholar
  46. 46.
    Wendland, H.: Fast evaluation of radial basis functions: methods based on partition of unity. In: Chui, C.K. (ed.) Approximation Theory X: Wavelets, Splines, and Applications, pp. 473–483. Vanderbilt University Press, Nashville (2002)Google Scholar
  47. 47.
    Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • S. De Marchi
    • 1
  • A. Martínez
    • 1
  • E. Perracchione
    • 1
    Email author
  • M. Rossini
    • 2
  1. 1.Dipartimento di Matematica, “Tullio Levi-Civita”Università di PadovaPaduaItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità di Milano - BicoccaMilanItaly

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