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  Convergence rates for exponentially ill-posed inverse problems with impulsive noise.

König, C., Werner, F., & Hohage, T. (2016). Convergence rates for exponentially ill-posed inverse problems with impulsive noise. SIAM Journal on Numerical Analysis, 54(1), 341-360. doi:10.1137/15M1022252.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-002A-06CF-0 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-002C-D0C6-4
Genre: Journal Article

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König, C.1, Author              
Werner, F.1, Author              
Hohage, T., Author
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1Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society, ou_1113580              

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Free keywords: variational regularization, impulsive noise, spaces of analytic functions
 Abstract: This paper is concerned with exponentially ill-posed operator equations with additive impulsive noise on the right-hand side, i.e., the noise is large on a small part of the domain and small or zero outside. It is well known that Tikhonov regularization with an $L^1$ data fidelity term outperforms Tikhonov regularization with an $L^2$ fidelity term in this case. This effect has recently been explained and quantified for the case of finitely smoothing operators. Here we extend this analysis to the case of infinitely smoothing forward operators under standard Sobolev smoothness assumptions on the solution, i.e., exponentially ill-posed inverse problems. It turns out that high order polynomial rates of convergence in the size of the support of large noise can be achieved rather than the poor logarithmic convergence rates typical for exponentially ill-posed problems. The main tools of our analysis are Banach spaces of analytic functions and interpolation-type inequalities for such spaces. We discuss two examples, the (periodic) backward heat equation and an inverse problem in gradiometry.

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Language(s): eng - English
 Dates: 2016-02-092016
 Publication Status: Published in print
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 Rev. Method: Peer
 Identifiers: DOI: 10.1137/15M1022252
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Title: SIAM Journal on Numerical Analysis
Source Genre: Journal
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Pages: - Volume / Issue: 54 (1) Sequence Number: - Start / End Page: 341 - 360 Identifier: -