Abstract
In this note, ill-posed problems are studied in the case when mapping A (not necessarily linear) is given only approximately. We prove the existence of three sequences: a sequence of operators \(\left( A_{n}\right) _{n}\), a sequence of arguments \(\left( x_{n}\right) _{n}\) , and a sequence of second members \(\left( u_{n}\right) _{n}\) converging on the exact values of the operator equation \(Ax=u\).
Similar content being viewed by others
References
Bangti J, Jun Z (2009) Augmented Tikhonov regularization. Inverse Probl 25:025001. https://doi.org/10.1088/0266-5611/25/2/025001
Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems. Kluwer, Dordrecht
Grasmair M (2011) Linear convergence rates for Tikhonov regularization with positively homogeneous functionals. Inverse Probl 27:075014. https://doi.org/10.1088/0266-5611/27/7/075014
Groetsc CW (1993) Inverse problems in the mathematical sciences. Vieweg, Braunschweig
Groetsch CW (1984) The theory of Tikhonov regularization for Fredholm equations of the first kind. Pitman, Boston
Hadamard J (1902) Sur les problèmes aux dérivées partielles et leur signification physique. Bull Univ Princeton 13:49–52
Hadamard J (1932) Le problème de cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, Paris
Jacobsen M (2000) Two-grid iterative methods for ill-posed problems. Technical University of Denmark, Lyngby
Läuter H, Leiro H (1997) Ill-posed problems and their optimal regularization, preprint 57. Humboldt-Universit ät, Berlin
Ramm AG (2002) Regularization of ill-posed problems with unbounded operators. J Math Anal Appl 271:547–550
Tikhonov AN, Arsenin VY (1977) Solution of ill-posed problems. Wiley, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benmeziane, R., Dahmani, A. Regularization of Ill-Posed Problems with Both Data and Operator are Perturbed. Iran J Sci Technol Trans Sci 43, 1157–1160 (2019). https://doi.org/10.1007/s40995-018-0563-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-018-0563-3