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MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction

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Advances in Trefftz Methods and Their Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 23))

Abstract

We investigate the application of the fading regularization method, in conjunction with the method of fundamental solutions, to the Cauchy problem in 2D anisotropic heat conduction. More precisely, we present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of over-prescribed noisy data taken on the remaining accessible boundary part. The accuracy, convergence, stability and robustness of the proposed numerical algorithm, as well as its capability to deblur noisy data, are validated by considering a test example in a 2D simply connected domain.

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References

  1. Cimetière, A., Delvare, F., Pons, F.: Une méthode inverse à régularisation évanescente. Comptes Rendus de l’Académie des Sciences - Série IIb - Mécanique 328, 639–644 (2000)

    MATH  Google Scholar 

  2. Cimetière, A., Delvare, F., Jaoua, M., Pons, F.: Solution of the Cauchy problem using iterated Tikhonov regularization. Inverse Probl. 17, 553–570 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Fairweather, G., Karageorghis, A., Martin, P.A.: The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Bound. Elem. 27, 759–769 (2003)

    Article  MathSciNet  Google Scholar 

  5. Gorzelańczyk, P., Kołodziej, J.A.: Some remarks concerning the shape of the shape contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng. Anal. Bound. Elem. 32, 64–75 (2008)

    Article  Google Scholar 

  6. Hadamard, J.: Lectures on Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)

    MATH  Google Scholar 

  7. Jin, B., Zheng, Y., Marin, L.: The method of fundamental solutions for inverse boundary value problems associated with the steady-state heat conduction in anisotropic media. Int. J. Numer. Meth. Eng. 65(11), 1865–1891 (2006)

    Article  MathSciNet  Google Scholar 

  8. Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Probl. Sci. Eng. 19, 309–336 (2011)

    Article  MathSciNet  Google Scholar 

  9. Keyhani, M., Polehn, R.A.: Finite difference modeling of anisotropic flows. J. Heat Transf. 117, 458–464 (1995)

    Article  Google Scholar 

  10. Kita, E., Kamiya, N.: Trefftz method: an overview. Adv. Eng. Softw. 24, 3–12 (1995)

    Article  Google Scholar 

  11. Knabner, P., Agermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York (2003)

    Google Scholar 

  12. Kozlov, V.A., Mazya, V.G., Fomin, A.V.: An iterative method for solving the Cauchy problem for elliptic equations. U.S.S.R. Comput. Math. Math. Phys. 31, 45–52 (1991)

    Google Scholar 

  13. Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. Comput. Math. Math. Phys. 4, 82–126 (1964)

    Article  MathSciNet  Google Scholar 

  14. Marin, L.: An alternating iterative MFS algorithm for the Cauchy problem in two-dimensional anisotropic heat conduction. Comput. Mater. Continua 12, 71–100 (2009)

    Google Scholar 

  15. Marin, L.: Stable boundary and internal data reconstruction in two-dimensional anisotropic heat conduction Cauchy problems using relaxation procedures for an iterative MFS algorithm. Comput. Mater. Continua 17(3), 233–274 (2010)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Mera, N.S., Elliott, L., Ingham, D.B., Lesnic, D.: The boundary element solution for the Cauchy steady heat conduction problem in an anisotropic medium. Int. J. Numer. Meth. Eng. 49, 481–499 (2000)

    Article  Google Scholar 

  18. Mera, N.S., Elliott, L., Ingham, D.B., Lesnic, D.: A comparison of boundary element formulations for steady state anisotropic heat conduction problems. Eng. Anal. Bound. Elem. 25, 115–128 (2001)

    Article  Google Scholar 

  19. Morozov, V.A.: On the solution of functional equations by the method of regularization. Doklady Math. 7, 414–417 (1966)

    MathSciNet  MATH  Google Scholar 

  20. Özişik, M.N.: Heat Conduction. Wiley, New York (1993)

    Google Scholar 

  21. Szabó, B., Babuška, I.: Introduction to Finite Element Analysis. Wiley, New York (2011)

    Book  Google Scholar 

Download references

Acknowledgements

This work was supported by a grant of Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN–III–P4–ID–PCE–2016–0083, within PNCDI III.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

    Liviu Marin

  2. Research Institute of the University of Bucharest (ICUB), University of Bucharest, Bucharest, Romania

    Liviu Marin

  3. Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest, Romania

    Liviu Marin

Authors
  1. Liviu Marin
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Correspondence to Liviu Marin .

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Editors and Affiliations

  1. Mathematics, CEMAT Instituto Superior Técnico, University of Lisbon, Lisboa, Portugal

    Carlos Alves

  2. Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus

    Andreas Karageorghis

  3. Instituto Superior Técnico, University of Lisbon, Lisboa, Portugal

    Vitor Leitão

  4. CEMAT Instituto Superior Técnico, University of Lisbon, Lisboa, Portugal

    Svilen Valtchev

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Marin, L. (2020). MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction. In: Alves, C., Karageorghis, A., Leitão, V., Valtchev, S. (eds) Advances in Trefftz Methods and Their Applications. SEMA SIMAI Springer Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-52804-1_7

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