Numerische Mathematik

, Volume 101, Issue 4, pp 643–662 | Cite as

Preconditioning Landweber iteration in Hilbert scales

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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−2 s 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 Open image in new window instead of Open image in new window 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.

Mathematics Subject Classification (2000)

65J15 65J20 65F10 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustria
  2. 2.Industrial Mathematics InstituteJohannes Kepler UniversityAustria

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