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ON THREE SPECTRAL REGULARIZATION METHODS FOR A BACKWARD HEAT CONDUCTION PROBLEM

  • Xiong, Xiang-Tuan ;
  • Fu, Chu-Li ;
  • Qian, Zhi
  • Published : 2007.11.30

Abstract

We introduce three spectral regularization methods for solving a backward heat conduction problem (BHCP). For the three spectral regularization methods, we give the stability error estimates with optimal order under an a-priori and an a-posteriori regularization parameter choice rule. Numerical results show that our theoretical results are effective.

Keywords

inverse problems;backward heat conduction;spectral regularization method;error estimate

References

  1. K. A. Ames, G. W. Clark, J. F. Epperson, and S. F. Oppenhermer, A comparison of regularizations for an ill-posed problem, Math. Comp. 67 (1998), no. 224, 1451–-1471 https://doi.org/10.1090/S0025-5718-98-01014-X
  2. L. Elden, F. Berntsson, and T. Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), no. 6, 2187–-2205 https://doi.org/10.1137/S1064827597331394
  3. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996
  4. L. C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998
  5. D. N. Hao, A mollification method for ill-posed problems, Numer. Math. 68 (1994), no. 4, 469–-506 https://doi.org/10.1007/s002110050073
  6. T. Hohage, Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim. 21 (2000), no. 3-4, 439–-464 https://doi.org/10.1080/01630560008816965
  7. M. Jourhmane and N. S. Mera, An iterative algorithm for the backward heat conduction problem based on variable relaxtion factors, Inverse Probl. Sci. Eng. 10 (2002), no. 4, 293–-308 https://doi.org/10.1080/10682760290004320
  8. S. M. Kirkup and M. Wadsworth, Solution of inverse diffusion problems by operatorsplitting methods, Applied Mathematical Modelling 26 (2002), no. 10, 1003–-1018 https://doi.org/10.1016/S0307-904X(02)00053-7
  9. R. Lattes and J. L. Lions, Methode de quasi-reversibilite et applications, Travaux et Recherches Mathematiques, No. 15 Dunod, Paris 1967
  10. N. S. Mera, L. Elliott, D. B. Ingham, and D. Lesnic, An iterative boundary element method for solving the one-dimensional backward heat conduction problem, International Journal of Heat and Mass Transfer 44 (2001), no. 10, 1937-1946 https://doi.org/10.1016/S0017-9310(00)00235-0
  11. N. S. Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl. Sci. Eng. 13 (2005), no. 1, 65-78 https://doi.org/10.1080/10682760410001710141
  12. K. Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for nonwell- posed problems, Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), pp. 161–176. Lecture Notes in Math., Vol. 316, Springer, Berlin, 1973
  13. R. E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974), 563-572 https://doi.org/10.1016/0022-247X(74)90008-0
  14. U. Tautenhahn, Optimal stable approximations for the sideways heat equation, J. Inverse Ill-Posed Probl. 5 (1997), no. 3, 287-307 https://doi.org/10.1515/jiip.1997.5.3.287
  15. U. Tautenhahn and T. Schroter, On optimal regularization methods for the backward heat equation, Z. Anal. Anwendungen 15 (1996), no. 2, 475-493 https://doi.org/10.4171/ZAA/711
  16. P. Mathe and S. Pereverzev, Geometry of ill-posed problems in variable Hilbert Scales, Inverse Problems 19 (2003), 789–-803 https://doi.org/10.1088/0266-5611/19/3/319
  17. U. Tautenhahn, Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim. 19 (1998), no. 3-4, 377–-398 https://doi.org/10.1080/01630569808816834

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