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The Role of Hilbert–Schmidt SVD basis in Hermite–Birkhoff interpolation in fractional sense

  • M. EsmaeilbeigiEmail author
  • O. Chatrabgoun
  • M. Cheraghi
Article
  • 30 Downloads

Abstract

The Hermite–Birkhoff (HB) interpolation is an extension of polynomial interpolation that appears when observation gives operational information. Additional capabilities in HB interpolation by considering fractional operators are created, as it occur in many applied systems. Due to the nice properties in kernel-based approximation, we intend to apply it to solve the HB interpolation in the fractional sense. We show the standard basis in kernel-based approximation is often insufficient for computing a stable solution in fractional HB interpolation. Because of the inherent ill-condition of kernel-based methods, we investigate the fractional HB interpolation using alternate Hilbert–Schmidt SVD (HS–SVD) bases, since it provides a linear transformation which can be applied analytically, and therefore, is able to remove a significant portion of the ill-conditioning. Also, the convergence and stability of the fractional HB interpolation using HS–SVD method are discussed. Numerical results show that in solving fractional HB interpolation, although the standard basis for many positive definite kernels is ill-conditioned in the flat limit, the HS–SVD basis solves the existing problem.

Keywords

Hermite–Birkhoff interpolation Fractional derivatives Hilbert–Schmidt SVD method Kernel-based approximation 

Mathematics Subject Classification

65D05 26A33 65M12 

Notes

References

  1. Allasia G, Cavoretto R, Rossi AD (2018) Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds. Appl. Math. Comput. 318(1):35–50MathSciNetGoogle Scholar
  2. Cavoretto R, Fasshauer GE, McCourt M (2015) An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels. Numer. Algorithms 68(2):393–422MathSciNetCrossRefGoogle Scholar
  3. Dell’Accio F, Tommaso FD (2016) Complete Hermite–Birkhoff interpolation on scattered data by combined Shepard operators. J. Comput. Appl. Math. 300:192–206MathSciNetCrossRefGoogle Scholar
  4. Esmaeilbeigi M, Chatrabgoun O, Cheraghi M (2018) Fractional Hermite interpolation using RBFs in high dimensions over irregular domains with application. J. Comput. Phys. 375:1091–1120MathSciNetCrossRefGoogle Scholar
  5. Farideh S, Habibollah S (2018) Discrete Hahn polynomials for numerical solution of two-dimensional variable-order fractional Rayleigh–Stokes problem. Comput. Appl. Math.  https://doi.org/10.1007/s40314-018-0631-5
  6. Fasshauer GE (1999) Solving differential equations with radial basis functions: Multilevel methods and smoothing. Adv. Comput. Math. 11:139–159MathSciNetCrossRefGoogle Scholar
  7. Fasshauer GE (2007) Meshfree approximation methods with Matlab, interdisciplinary mathematical sciences, vol 6. World Scientific Publishing, SingaporeCrossRefGoogle Scholar
  8. Fasshauer GE, McCourt M (2012) Stable evaluation of Gaussian RBF interpolants. SIAM J. Sci. Comput. 34(2):737–762MathSciNetCrossRefGoogle Scholar
  9. Fasshauer GE, McCourt M (2015) Kernel-based approximation methods using MATLAB, interdisciplinary mathematical sciences, vol 19. World Scientific Publishing, SingaporezbMATHGoogle Scholar
  10. Goldman G (2017) A case of multivariate Birkhoff interpolation using high order derivatives. arXiv:1603.04045 [math. CA]
  11. Khader MM, El Danaf TS, Hendy AS (2013) A computational matrix method for solving systems of high order fractional differential equations. Appl. Math. Model. 37(6):4035–4050MathSciNetCrossRefGoogle Scholar
  12. Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations, North-Holland mathematics studies, vol 204. Elsevier, AmsterdamGoogle Scholar
  13. Manh PV (2017) Hermite interpolation with symmetric polynomials. Numer. Algorithms 76(3):709–725MathSciNetCrossRefGoogle Scholar
  14. Michael M, Fasshauer GE (2017) Stable likelihood computation for Gaussian random fields. Springer, Berlin, pp 17–943.  https://doi.org/10.1007/978-3-319-55556-0-16
  15. Quang MH, Niyogi P, Yao Y (2006) Mercer’s Theorem, feature maps, and smoothing. In: International Conference on Computational Learning Theory (COLT), pp 154–168Google Scholar
  16. Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high-frequency financial data, an empirical study. Phys. Stat. Mech. Appl. 314:749–755CrossRefGoogle Scholar
  17. Raissi M, Perdikaris P, Karniadakisa GE (2017) Machine learning of linear differential equations using Gaussian processes. J. Comput. Phys. 348(1):683–693MathSciNetCrossRefGoogle Scholar
  18. Rashidiniaa J, Fasshauer GE, Khasi M (2016) A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems. Comput. Math. Appl. 72(1):178–193MathSciNetCrossRefGoogle Scholar
  19. Rashidiniaa J, Khasia M, Fasshauer GE (2018) A stable Gaussian radial basis function method for solving nonlinear unsteady convection-diffusion-reaction equations. Comput. Math. Appl. 75(5):1831–1850MathSciNetCrossRefGoogle Scholar
  20. Sayevand K, Machado JT, Moradi V (2018) A new non-standard finite difference method for analyzing the fractional Navier–Stokes equations. Comput. Math. Appl.  https://doi.org/10.1016/j.camwa.2018.12.016
  21. Wendland H (2005) Scattered data approximation, vol 17. Cambridge monographs on applied and computational mathematics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  22. Yue C, Baoli Y, Yang L, Hong L (2018) Crank–Nicolson WSGI difference scheme with finite element method for multi-dimensional time-fractional wave problem. Comput. Appl. Math.  https://doi.org/10.1007/s40314-018-0626-2

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical Sciences and StatisticsMalayer UniversityMalayerIran
  2. 2.Department of Statistics, Faculty of Mathematical Sciences and StatisticsMalayer UniversityMalayerIran

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