Abstract
In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems. As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case s≤0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the Hilbert scale operator L−2s appearing in the iteration acts as a preconditioner, which significantly reduces the number of iterations needed to match an appropriate stopping criterion. Additionally, we carry out our analysis under significantly relaxed conditions, i.e., we only require 
instead of 
which is the usual condition for regularization in Hilbert scales. The assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones.
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References
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problem. Kluwer Academic Publishers, 1996
Fridman, V.: Methods of successive approximations for Fredholm integral equations of the first kind. Usp. Mat. Nauk 11, 233–234 (1956) (in Russian)
Gorenflo, R., Vesella, S.: Abel Integral Equations: Analysis and Applications. Number 1461 in Lecture Notes in Math. Springer, Berlin, 1991
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston, 1984
Hanke, M.: Accelerated Landweber iterations for the solution of ill-posed equations. Numer. Math. 60, 341–373 (1991)
Hanke, M., Neubauer, A., Scherzer, O.: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995)
Hohage, T.: Iterative Methods in Inverse Obstacle Scattering: Regularization Theory of Linear and Nonlinear Exponentially Ill-Posed Problems. PhD thesis, University of Linz, 1999
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Problems. 2005. (in preparation)
Krein, S.G., Petunin, J.I.: Scales of Banach spaces. Russ. Math. Surv. 21, 85–160 (1966)
Kreß, R.: Linear Integral Equations. Springer, Berlin, 1989
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications: Volume I. Springer, Berlin - Heidelberg, 1972
Natterer, F.: Error bounds for Tikhonov regularization in Hilbert scales. Appl. Anal. 18, 29–37 (1984)
Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart, 1986
Neubauer, A.: Tikhonov regularization of nonlinear ill-posed problems in Hilbert scales. Appl. Anal. 46, 59–72 (1992)
Neubauer, A.: On Landweber iteration for nonlinear ill-posed problems in Hilbert scales. Numer. Math. 85, 309–328 (2000)
Tautenhahn, U.: Error estimates for regularization methods in Hilbert scales. SIAM J. Numer. Anal. 33, 2120–2130 (1996)
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supported by the Austrian Science Foundation (FWF) under grant SFB/F013
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Egger, H., Neubauer, A. Preconditioning Landweber iteration in Hilbert scales. Numer. Math. 101, 643–662 (2005). https://doi.org/10.1007/s00211-005-0622-5
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DOI: https://doi.org/10.1007/s00211-005-0622-5