Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

REGULARIZED SOLUTION TO THE FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND WITH NOISY DATA

  • Wen, Jin (School of Mathematics and Statistics, Lanzhou University) ;
  • Wei, Ting (School of Mathematics and Statistics, Lanzhou University)
  • Received : 2010.04.15
  • Accepted : 2010.06.22
  • Published : 2011.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we use a modified Tikhonov regularization method to solve the Fredholm integral equation of the first kind. Under the assumption that measured data are contaminated with deterministic errors, we give two error estimates. The convergence rates can be obtained under the suitable choices of regularization parameters and the number of measured points. Some numerical experiments show that the proposed method is effective and stable.

Keywords

References

  1. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68(1950), 337-404. https://doi.org/10.1090/S0002-9947-1950-0051437-7
  2. J. Cullum, Numerical differentiation and regularization, SIAM J. Numer. Anal. 8(1971), 254-265. https://doi.org/10.1137/0708026
  3. J.N. Franklin, On Tikhonov's method for ill-posed problems, Math. Comp. 28(1974), 889-907.
  4. C.W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, volume 105 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1984.
  5. C.W. Groetsch, Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind, Integral Equations Operator Theory 13(1)(1990), 67-75. https://doi.org/10.1007/BF01195293
  6. J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale University Press, New Haven, CT, U.S.A., 1923.
  7. J.W. Hilgers, On the equivalence of regularization and certain reproducing kernel Hilbert space approaches for solving first kind problems, SIAM J. Numer. Anal. 13(2)(1976), 172-184. https://doi.org/10.1137/0713018
  8. R. Kress, Linear integral equations, volume 82 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1989.
  9. G. Li and M.Z. Nashed, A modified Tikhonov regularization for linear operator equations, Numer. Funct. Anal. Optim. 26(4-5)(2005), 543-563. https://doi.org/10.1080/01630560500248389
  10. M.A. LukasConvergence rates for regularized solutions, Math. Comp. 51(183)(1988), 107-131. https://doi.org/10.1090/S0025-5718-1988-0942146-8
  11. M.A. Lukas, Convergence rates for moment collocation solutions of linear operator equations, Numer. Funct. Anal. Optim. 16(5-6)(1995), 743-750. https://doi.org/10.1080/01630569508816641
  12. M.Z. Nashed and G. Wahba, Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind, Math. Comp. 28(1974), 69-80. https://doi.org/10.1090/S0025-5718-1974-0461895-1
  13. M.Z. Nashed and G. Wahba, Generalized inverses in reproducing kernel spaces: an approach to regularization of linear operator equations, SIAM J. Math. Anal. 5(1974), 974-987. https://doi.org/10.1137/0505095
  14. A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977.
  15. G.Wahba, A class of approximate solutions to linear operator equations, J. Approximation Theory 9(1973), 61-77. https://doi.org/10.1016/0021-9045(73)90112-3
  16. G. Wahba, Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind, J. Approximation Theory 7(1973), 167-185. https://doi.org/10.1016/0021-9045(73)90064-6
  17. G. Wahba, Practical approximate solutions to linear operator equations when the data are noisy, SIAM Journal on Numerical Analysis 14(4)(1977), 651-667. https://doi.org/10.1137/0714044
  18. G. Wahba, Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.