Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the Cauchy problem of Laplace’s equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alessandrini, G.: Stable determination of a crack from boundary measurements. Proc. R. Soc. Edinb. Sect. A 123(3), 497–516 (1993)

    MATH  MathSciNet  Google Scholar 

  2. Ang, D.D., Nghia, H., Tam, N.C.: Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer: a three-dimensional model. Acta Math. Vietnam. 23(1), 65–74 (1998)

    MATH  MathSciNet  Google Scholar 

  3. Berntsson, F., Eldén, L.: Numerical solution of a Cauchy problem for the Laplace equation. Inverse Probl. 17(4), 839–853 (2001) [Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000)]

    Article  MATH  Google Scholar 

  4. Bourgeois, L.: A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Probl. 21(3), 1087–1104 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheng, J., Hon, Y.C., Wei, T., Yamamoto, M.: Numerical computation of a Cauchy problem for Laplace’s equation. ZAMM Angew Z. Math. Mech. 81(10), 665–674 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davis, P.J.: Circulant Matrices. Wiley-Interscience, New York (1979)

    MATH  Google Scholar 

  7. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9(1–2), 69–95 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Falk, R.S., Monk, P.B.: Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation. Math. Comp. 47(175), 135–149 (1986)

    MATH  MathSciNet  Google Scholar 

  9. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Boundary Integral Methods: Numerical and Mathematical Aspects, vol. 1 of Comput. Eng., pp. 103–176. WIT/Comput. Mech. Publ., Boston (1999)

    Google Scholar 

  10. Hadamard, J.: Lectures on Cauchy Problems in Linear Partial Differential Equations. Yale University Press, New Heaven (1923)

    Google Scholar 

  11. Hào, D.N., Lesnic, D.: The Cauchy problem for Laplace’s equation via the conjugate gradient method. IMA Appl. J. Math. 65(2), 199–217 (2000)

    Article  MATH  Google Scholar 

  12. Hon, Y.C., Wei, T.: A fundamental solution method for inverse heat conduction problem. Eng. Anal. Bound. Elem. 28, 489–495 (2004)

    Article  MATH  Google Scholar 

  13. Hon, Y.C., Wei, T.: The method of fundamental solution for solving multidimensional inverse heat conduction problems. CMES Comput. Model. Eng. Sci. 7(2), 119–132 (2005)

    MATH  MathSciNet  Google Scholar 

  14. Jin, B., Zheng, Y.: A meshless method for some inverse problems associted with the Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 195(19–22), 2270–2280 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Karageorghis, A., Fairweather, G.: The method of fundamental solutions for the numerical solution of the biharmonic equation. Comput. J. Phys. 69, 433–459 (1987)

    MathSciNet  Google Scholar 

  16. Katsurada, M.: A mathematical study of the charge simulation method. II. Fac. J. Sci. Univ. Tokyo Sect. IA Math. 36(1), 135–162 (1989)

    MATH  MathSciNet  Google Scholar 

  17. Katsurada, M., Okamoto, H.: A mathematical study of the charge simulation method. I. Fac. J. Sci. Univ. Tokyo Sect. IA Math. 35(3), 507–518 (1988)

    MATH  MathSciNet  Google Scholar 

  18. Katsurada, M., Okamoto, H.: The collocation points of the fundamental solution method for the potential problem. Comput. Math. Appl. 31(1), 123–137 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Klibanov, M.V., Santosa, F.: A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM Appl. J. Math. 51(6), 1653–1675 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kondapalli, P.S., Shippy, D.J., Fairweather, G.: Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions. Acoust. J. Soc. Am. 91(4 Pt 1), 1844–1854 (1992)

    Article  Google Scholar 

  21. Kozlov, V.A., Maz′ya, V.G., Fomin, A.V.: An iterative method for solving the Cauchy problem for elliptic equations. Zh. Vychisl. Mat. Mat. Fiz. 31(1), 64–74 (1991)

    MATH  MathSciNet  Google Scholar 

  22. Kythe, P.K.: Fundamental Solutions for Differential Operators and Applications. Birkhäuser Boston, Boston (1996)

    MATH  Google Scholar 

  23. Marin, L.: A meshless method for solving the Cauchy problem in three-dimensional elastostatics. Comput. Math. Appl. 50(1–2), 73–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Marin, L.: A meshless method for the numerical solution of Cauchy problem associated with three-dimensional Helmholtz-type equations. Appl. Math. Comput. 165, 355–374 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Marin, L., Lesnic, D.: The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation. Math. Comput. Modelling 42(3–4), 261–278 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Marin, L., Lesnic, D.: The method of fundmental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations. Comput. Struct. 83, 267–278 (2005)

    Article  MathSciNet  Google Scholar 

  27. Mathon, R., Johnston, R.L.: The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14(4), 638–650 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  28. Murota, K.: Comparison of conventional and “invariant” schemes of fundamental solutions method for annular domains. Jpn. J. Ind. Appl. Math. 12, 61–85 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Ohe, T., Ohnaka, K.: Uniqueness and convergence of numerical solution of the Cauchy problem for the Laplace equation by a charge simulation method. Jpn. J. Ind. Appl. Math. 21(3), 339–359 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. Poullikkas, A., Karageorghis, A., Georgiou, G.: The method of fundamental solutions for three-dimensional elastostatics problems. Comput. Struct. 80(3–4), 365–370 (2002)

    Article  MathSciNet  Google Scholar 

  31. Reinhardt, H.J., Han, H., Hào D.N.: Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation. SIAM J. Numer. Anal. 36(3), 890–905 (1999; electronic)

    Article  MathSciNet  Google Scholar 

  32. Smyrlis, Y.S., Karageorghis, A.: Some aspects of the method of fundamental solutions for certain biharmonic problems. CMES Comput. Model. Eng. Sci. 4(5), 535–550 (2003)

    MATH  MathSciNet  Google Scholar 

  33. Wang, Y., Rudy, Y.: Application of the method of fundamental solutions to potential-based inverse electrocardiography. Ann. Biomed. Eng. 34(8), 1272–1288 (2006)

    Article  Google Scholar 

  34. Wei, T., Hon, Y.C., Ling, L.: Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Eng. Anal. Bound. Elem. 31(4), 373–385 (2007)

    Article  MATH  Google Scholar 

  35. Zhou, D.Y., Wei, T.: The method of fundamental solutions for solving a Cauchy problem of Laplace’s equation in a multi-connected domain. Inverse Probl. Sci. Eng. 16(3), 389–411 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, People’s Republic of China

    T. Wei & D. Y. Zhou

Authors
  1. T. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Y. Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Wei.

Additional information

Communicated by the guest editors Benny Hon, Jin Cheng and Masahiro Yamamoto.

The work described in this paper was supported by the NSF of China (10571079, 10671085) and the program of NCET.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wei, T., Zhou, D.Y. Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions. Adv Comput Math 33, 491–510 (2010). https://doi.org/10.1007/s10444-009-9134-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-009-9134-7

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation