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Convergence rates for inverse problems with impulsive noise.

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Werner,  F.
Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society;

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Hohage, T., & Werner, F. (2014). Convergence rates for inverse problems with impulsive noise. SIAM Journal on Numerical Analysis, 52(3), 1203-1221. doi:10.1137/130932661.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0019-7ECD-2
Abstract
We study inverse problems $F(f) =g$ with perturbed right-hand side $g^{\rm obs}$ corrupted by so-called impulsive noise, i.e., noise which is concentrated on a small subset of the domain of definition of $g$. It is well known that Tikhonov-type regularization with an $\mathbf{L}^1$ data fidelity term yields significantly more accurate results than Tikhonov regularization with classical $\mathbf{L}^2$ data fidelity terms for this type of noise. The purpose of this paper is to provide a convergence analysis explaining this remarkable difference in accuracy. Our error estimates significantly improve previous error estimates for Tikhonov regularization with $\mathbf{L}^1$-fidelity term in the case of impulsive noise. We present numerical results which are in good agreement with the predictions of our analysis.