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Discrepancy Principle and Convergence Rates in Regularization of Monotone Ill-Posed Problems

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Abstract

The convergence rates of the regularized solution as well as its Galerkin approximations for nonlinear monotone ill-posed problems in a Banach space are established on the basis of the choice of a regularization parameter by the Morozov discrepancy principle.

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REFERENCES

  1. I. P. Ryazantseva, “On one algorithm for the solution of nonlinear monotone equations with unknown error estimate for initial data,” Zh. Vychisl. Mat. Mat. Fiz., 29, No. 10, 1572–1576 (1989).

    Google Scholar 

  2. Ya. I. Al'ber, “On the solution of nonlinear equations with monotone operators in a Banach space,” Sib. Mat. Zh., 16, No. 1, 3–11 (1975).

    Google Scholar 

  3. Ya. I. Al'ber and I. P. Ryazantseva, “ On the solution of nonlinear problems with monotone discontinuous mappings,” Dyfferents. Uravn., 15, No. 2, 331–342 (1979).

    Google Scholar 

  4. I. P. Ryazantseva, “On the construction of regularization algorithms for equations with monotone mappings,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 39–43 (1981).

  5. I. P. Ryazantseva, “On the Galerkin method for the solution of equations with discontinuous monotone operators,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 7, 68–72 (1978).

  6. H. W. Engl, “Discrepancy principle for Tikhonov regularization of ill-posed problems leading to optimal convergence rates,” J. Optimization Theory Appl., 52, 209–215 ( 1987).

    Google Scholar 

  7. C. W. Groetsch, “On the asymptotic order of accuracy of Tikhonov regularization,” J. Optimization Theory Appl., 41, 293–298 (1983).

    Google Scholar 

  8. J. Guacaneme, “On the optimal convergence for regularized Tikhonov approximations,” J. Optimization Theory Appl., 58, 127–131 (1988).

    Google Scholar 

  9. E. Schock, “On the asymptotic order of accuracy of Tikhonov regularization,” J. Optimization Theory Appl., 44, 95–104 (1984).

    Google Scholar 

  10. E. Schock, “Parameter choice by discrepancy principle for the Tikhonov regularization of ill-posed problems,” Integral Equations Operator Theory, 7, 895–899 (1984).

    Google Scholar 

  11. A. Neubauer, “An a-posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates,” SIAM J. Numer. Math., 25, 1313–1326 (1988).

    Google Scholar 

  12. H. W. Engl, K. Kunish, and A. Neubauer, “Convergence rates for Tikhonov regularization of nonlinear ill-posed problems,” Inverse Problems, 5, 523–540 (1989).

    Google Scholar 

  13. Q. N. Jin, “Application of the modified discrepancy principle to Tikhonov regularization of nonlinear ill-posed problems,” SIAM J. Numer. Anal., 36, No. 2, 475–490 (1999).

    Google Scholar 

  14. Q. N. Jin, “On a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems,” Numer. Math., 83, 139–159 (1999).

    Google Scholar 

  15. O. Scherzer, “The use of Morozov's discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems,” Computing, 51, 45–60 (1993).

    Google Scholar 

  16. O. Scherzer, H. W. Engl, and K. Kunish, “Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems,” SIAM J. Numer. Math., 30, 1796–1838 (1999).

    Google Scholar 

  17. Ng. Buong, Convergence Rates and Finite-Dimensional Approximations of Nonlinear Ill-Posed Problem Involving Monotone Operator in Banach Space, Preprint No. 385, ICTP, Triest (1992).

  18. Ng. Buong, “On ill-posed problems in Banach spaces,” Bull. South-East Math. J., 21, 95–103 (1997).

    Google Scholar 

  19. B. Hofmann and O. Scherzer, “Factor influencing the ill-posedness of nonlinear problems,” Inverse Problems, 10, 1277–1297 (1994).

    Google Scholar 

  20. M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations [in Russian], Nauka, Moscow (1972).

    Google Scholar 

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Nguyen Buong Discrepancy Principle and Convergence Rates in Regularization of Monotone Ill-Posed Problems. Ukrainian Mathematical Journal 55, 1198–1206 (2003). https://doi.org/10.1023/B:UKMA.0000010616.85298.fe

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