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An Adaptive LOOCV-Based Algorithm for Solving Elliptic PDEs via RBF Collocation

  • R. CavorettoEmail author
  • A. De Rossi
Conference paper
  • 75 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

Abstract

We present a new adaptive scheme for solving elliptic partial differential equations (PDEs) through a radial basis function (RBF) collocation method. Our adaptive algorithm is meshless and it is characterized by the use of an error indicator, which depends on a leave-one-out cross validation (LOOCV) technique. This approach allows us to locate the areas that need to be refined, also including the chance to add or remove adaptively any points. The algorithm turns out to be flexible and effective by means of a good interaction between error indicator and refinement procedure. Numerical experiments point out the performance of our scheme.

Keywords

Meshfree methods Adaptive algorithms Refinement techniques Poisson problems 

References

  1. 1.
    Cavoretto, R., De Rossi, A.: Adaptive meshless refinement schemes for RBF-PUM collocation. Appl. Math. Lett. 90, 131–138 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, W., Fu, Z.-J., Chen, C.S.: Recent Advances on Radial Basis Function Collocation Methods. Springer Briefs in Applied Science and Technology. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-642-39572-7CrossRefzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Co., Singapore (2007)CrossRefGoogle Scholar
  5. 5.
    Hon, Y.C., Schaback, R.: On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119, 177–186 (2001)MathSciNetzbMATHGoogle Scholar
  6. 6.
    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, 147–161 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Lee, C.-F., Ling, L., Schaback, R.: On convergent numerical algorithms for unsymmetric collocation. Adv. Comput. Math. 30, 339–354 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Oanh, D.T., Davydov, O., Phu, H.X.: Adaptive RBF-FD method for elliptic problems with point singularities in 2D. Appl. Math. Comput. 313, 474–497 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rippa, S.: An algorithm for selecting a good value for the parameter \(c\) in radial basis function interpolation. Adv. Comput. Math. 11, 193–210 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)Google Scholar
  12. 12.
    Yang, J., Liu, X., Wen, P.H.: The local Kansa’s method for solving Berger equation. Eng. Anal. Bound. Elem. 57, 16–22 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics “Giuseppe Peano”University of TorinoTorinoItaly

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