Abstract
In this paper, iterative regularization of Quasi-Newton methods by spectral cutoff is investigated. The proposed Iteratively Truncated Newton’s (ITN) scheme can be used for solving nonlinear irregular operator equations or unstable minimization problems. This algorithm is, in fact, a special case of a general procedure developed in Bakushinsky and Kokurin (Iterative methods for Ill-Posed operator equations with smooth operators, Springer, Great Britain, 2004). However convergence and stability analysis conducted in Bakushinsky and Kokurin is not applicable here since the generating function is not analytic. Therefore, the paper presents an independent study of ITN method, which is carried out under the source condition that is the weakest possible if no restrictions on the structure of the operator are imposed. As a practical example, a 2D nonlinear inverse magnetometry problem (Akimova and Vasin Stable parallel algorithms for solving the inverse gravimetry and magnitimetry problems, CD Proceedings of the 9th International Conference on Numerical Methods in Continuum Mechanics, Zilina, Slovakia, September 9–12 (2003); Vasin et al. Methods for solving an inverse magnetometry problem. (Russian) Sib. Èlektron. Mat. Izv. 5 620–631 (2008); Vasin and Skorik Iterative processes of gradient type with applications to gravimetry and magnetometry inverse problems. J. Inverse Ill-Posed Probl. 18, N8, 855–876 (2010)) is considered to illustrate advantages and limitations of the proposed algorithm.
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The work is supported by NSF grant (DMS-1112897).
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Bakushinsky, A.B., Smirnova, A.B., Liu, H. (2013). Theoretical and Numerical Study of Iteratively Truncated Newton’s Algorithm. In: Beilina, L. (eds) Applied Inverse Problems. Springer Proceedings in Mathematics & Statistics, vol 48. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7816-4_1
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