Mathematics – Functional Analysis
Scientific paper
2008-06-19
Inverse Problems 24(5) id. 055020 (2008)
Mathematics
Functional Analysis
15 pages
Scientific paper
10.1088/0266-5611/24/5/055020
We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$.
Grasmair Markus
Haltmeier Markus
Scherzer Otmar
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