Journal of Inverse and Ill-posed Problems
Editor-in-Chief: Kabanikhin, Sergey I.
6 Issues per year
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Rank 59 out of 312 in category Mathematics and 93 out of 254 in Applied Mathematics in the 2015 Thomson Reuters Journal Citation Report/Science Edition
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Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
- Zentrum für Technomathematik, Fachbereich 3, Universität Bremen, PO Box 330440, 28334 Bremen, Germany. Email: email@example.com
This paper addresses the regularization by sparsity constraints by means of weighted ℓp penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed.
For p < 1 only preliminary results are shown. These results indicate that, different from p ≥ 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.
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