On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation

Abstract

In the context of numerical approximation of partial differential equations, meshless methods are recent developments, which have been reported to be relatively straightforward, yet provide better convergence and accuracy as compared to the conventional mesh-based approaches, for some specific problems like stress–strain analysis and modeling in a deforming media. Among several proposed schemes, strong-form collocation schemes using radial basis functions are easy-to-program, require no mesh in the domain or at the boundary, avoid numerical integration, and have similar formulations for all dimensions. Some of the radial basiss functions like the Gaussian, multiquadric and inverse-multiquadric are infinitely smooth. The standard global meshless formulations based on such radial basis functions are often found to lead towards ill-conditioned systems of linear equations. For such formulations, we investigate the degree of ill-conditioning against degrees of freedom for different radial basis functions. Also, the convergence of four iterative methods and pseudoinverse approaches has been tested for the solution of such ill-conditioned systems. In order to compute the pseudoinverse, two different approaches, i.e., singular value decomposition and full rank Cholesky factorization have been used. A set of numerical experiments, performed here, demonstrate that pseudoinverse approach based on singular value decomposition performs better than the iterative methods. Although, pseudoinverse computation via full rank Cholesky factorization is faster, it is not recommended in global meshless collocation schemes due to poor convergence at relatively large degrees of freedom.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Beatson, R., Cherrie, J., Mouat, C.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11(2–3), 253–270 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Beatson, R.K., Light, W.A., Billings, S.: Fast solution of the radial basis function interpolation equations: domain decomposition methods. SIAM J. Sci. Comput. 22(5), 1717–1740 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Chen, W., Fu, Z., Chen, C.: Recent Advances in Radial Basis Function Collocation Methods. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  4. 4.

    Courrieu, P.: Fast computation of Moore–Penrose inverse matrices. Neural Inf. Process. Lett. Rev. 8(2), 25–29 (2005)

    Google Scholar 

  5. 5.

    Dehghan, M., Shokri, A.: A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions. Comput. Math. Appl. 54(1), 136–146 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Demirkaya, G., Wafo, S.C., Ilegbusi, O.: Direct solution of Navier–Stokes equations by radial basis functions. Appl. Math. Model. 32(9), 1848–1858 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Fasshauer, G., McCourt, M.: The uncertainty principle—an unfortunate misconception. In: Kernel-Based Approximation Methods Using MATLAB, pp. 199–201 (2016)

  8. 8.

    Fasshauer, G.F.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Inc., River Edge (2007)

    Book  MATH  Google Scholar 

  9. 9.

    Ferreira, A., Roque, C., Jorge, R.: Static and free vibration analysis of composite shells by radial basis functions. Eng. Anal. Bound. Elem. 30(9), 719–733 (2006)

    Article  MATH  Google Scholar 

  10. 10.

    Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869–892 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    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)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Franke, R.: A Critical Comparison of Some Methods for Interpolation of Scattered Data. Defense Technical Information Center (1979)

  13. 13.

    Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6(1), 1–35 (1994)

  14. 14.

    Hardy, Rolland L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76(8), 2156–2202 (1971)

    Article  Google Scholar 

  15. 15.

    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)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Kansa, E.J., Hon, Y.C.: Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appl. 39(7–8), 123–137 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Kindelan, M., Bernal, F., González-Rodríguez, P., Moscoso, M.: Application of the RBF meshless method to the solution of the radiative transport equation. J. Comput. Phys. 229(5), 1897–1908 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Ling, L., Kansa, E.J.: Preconditioning for radial basis functions with domain decomposition methods. Math. Comput. Model. 40(13), 1413–1427 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26(2), 394–395 (1920)

    Google Scholar 

  20. 20.

    Penrose, R.: A generalized inverse for matrices. Math. Proc. Camb. Philos. Soc. 51(3), 406–413 (1955)

    Article  MATH  Google Scholar 

  21. 21.

    Sajavicius, S.: Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions—a case of two-dimensional Poisson equation. Eng. Anal. Bound. Elem. 37(4), 788–804 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Sarler, B.: Towards a mesh-free computation of transport phenomena. Eng. Anal. Bound. Elem. 26(9), 731–738 (2002)

    Article  MATH  Google Scholar 

  23. 23.

    Sarler, B.: From global to local radial basis function collocation method for transport phenomena. In: Leitão, V., Alves, C., Armando Duarte, C. (eds.) Advances in Meshfree Techniques, Computational Methods in Applied Sciences, vol. 5, pp. 257–282. Springer, Dordrecht (2007)

    Chapter  Google Scholar 

  24. 24.

    Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Strang, G.: Computational Science and Engineering. Wellesley-Cambridge Press, Wellesley (2007)

    MATH  Google Scholar 

  26. 26.

    ul Islam, S., Aziz, I., Ahmad, M.: Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions. Comput. Math. Appl. 69(3), 180–205 (2015)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sankar K. Nath.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mishra, P.K., Nath, S.K. On the Convergence of Iterative Methods and Pseudoinverse Approaches in Global Meshless Collocation. Int. J. Appl. Comput. Math 3, 3987–4000 (2017). https://doi.org/10.1007/s40819-016-0272-6

Download citation

Keywords

  • Radial basis function
  • Ill-conditioning
  • Generalized inverse
  • Iterative methods
  • Meshless methods

Mathematics Subject Classification

  • 65
  • 68