Tikhonov regularization of nonlinear ill-posed problems with closed operators in Hilbert scales

Journal of Inverse and Ill-Posed Problems

Volumn 5, Issue 4, 1997, Pages 363-376

  Liu, F ^a   Nashed, M Z ^b  

^a DELAWARE STATE UNIVERSITY   (United States)

^b UNIVERSITY OF DELAWARE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042960225     PISSN: 09280219     EISSN: None     Source Type: Journal    
DOI: 10.1515/jiip.1997.5.4.363     Document Type: Article

References (30)

1. A.B. Bakushinskiĭ and A.V. Goncharskiĭ, Ill-Posed Problems: Theory and Applications. Mathematics and Its Applications, Vol. 301. Kluwer Academic Publishers, Dordrecht, 1994.
2. X. Chen, Z. Nashed and L. Qi, Convergence of Newton's method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim. (1997) 7, No. 2, 445–462.
3. F. Colonius and K. Kunisch, Stability for parameter estimation in two-point boundary value problems. J. Reine Angew. Math. (1986) 370, 1–29.
4. H.W. Engl, Regularization methods for the stable solution of inverse problems. Surv. Math. Ind. (1993) 3, No.2, 71–143.
5. H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.
6. H.W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Problems (1989) 5, No. 4, 523–540.
7. H.W. Engl and A. Neubauer, Convergence rates for Tikhonov regularization in finitedimensional subspaces of Hilbert scales. Proc. Amer. Math. Soc. (1988) 102, No.3, 587–592.
8. C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind. Pitman, Boston, 1984.
9. S.G. Krein and YU. I. Petunin, Scales of Banach spaces. Russian Math. Surveys (1966) 21, 85–160.
10. K. Kunisch and W. Ring, Regularization of nonlinear ill-posed problems with closed operators. Numer. Funct. Anal. Optimiz. (1993) 14, No. 3-4, 389–404.
11. F. Liu, Ill-posed nonlinear operator equations and monotone variational inequalities. Ph. D. Dissertation, University of Delaware, Delaware, 1995.
12. F. Liu and M. Z. Nashed, Convergence of regularized solutions of nonlinear ill-posed problems with monotone operators. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Applied Mathematics, Vol. 177. Marcel Dekker, New York, 1996, 353–361.
13. J. Locker and P.M. Prenter, Regularization with differential operators. I. General theory. J. Math. Anal Appl. (1980) 74, No.2, 504–529.
14. V.A. Morozov, Methods for Solving Incorrectly Posed Problems. Springer-Verlag, New York-Berlin, 1984.
15. V.A. Morozov, Regularization Methods for Ill-Posed Problems. CRC Press, Boca Raton, FL, 1993.
16. M.Z. Nashed, Approximate regularized solutions to improperly posed linear integral and operator equations. In: Constructive and Computational Methods for Differential and Integral Equations. Symposium, Indiana Univ., February 17–20, 1974 (Eds. D.L. Colton and R. P. Gilbert). Lecture Notes in Mathematics, Vol. 430. Springer-Verlag, Berlin-Heidelberg-New York, 1974, 289–332.
17. M.Z. Nashed, On nonlinear ill-posed problems. I. Classes of operator equations and minimization of functionals. In: Nonlinear Analysis and Applications (Arlington, Tex., 1986). Lecture Notes in Pure and Applied Mathematics, Vol. 109. Marcel Dekker, New York, 1987, 351–373.
18. M.Z. Nashed and X. J. Chen, Convergence of Newton-like methods for singular operator equations using outer inverses. Numer. Math. (1993) 66, No. 2, 235–257.
19. M.Z. Nashed and F. Liu, On nonlinear ill-posed problems. II. Monotone operator equations and monotone variational inequalities. In: Theory and Applications of Non-Linear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Applied Mathematics, Vol. 178. Marcel Dekker, New York, 1996, 223–240.
20. F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales. Applicable Analysis (1984) 18, No. 1–2, 29–37.
21. A. Neubauer. An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates. SIAM J. Numer. Anal. (1988) 25, No. 6, 1313-1326.
22. A. Neubauer, Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Applicable Analysis (1992) 46, No. 1–2, 59–72.
23. T. Schröter and U. Tautenhahn, Error estimates for Tikhonov regularization in Hilbert scales. Numer. Funct. Anal. Optimiz. (1994) 15, No. 1-2, 155–168.
24. T.I. Seidman, Convergent approximation methods for ill-posed problems. Part I: General theory. Part II: Applications. Control and Cybernetics (1981) 10, No. 1-2, 31-49, 51-71.
25. T.I. Seidman and C.R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems. Inverse Problems (1989) 5, No.2. 227–238.
26. U. Tautenhahn, Error estimates for regularized solutions of nonlinear ill-posed problems. Inverse Problems (1994) 10, No. 2, 485–500.
27. A.N. Tikhonov, On the regularization of ill-posed problems. Dokl. Akad. Nauk SSSR (1963) 153, 49–52 (in Russian).
28. A. N. Tikhonov, A. V. Goncharskiĭ, V. V. Stephanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publishers, Dordrecht, 1995.
29. V.V. Vasin and A. L. Ageev, Ill-Posed Problems with a Priori Information. VSP, Utrecht, The Netherlands, 1995.
30. C. R. Vogel, A constrained least squares regularization method for nonlinear ill-posed problems. SIAM J. Control and Optimization (1990) 28, No.1, 34–49.