A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators
Autor: | Barbara Kaltenbacher, Bernd Hofmann, Otmar Scherzer, C Pöschl |
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Rok vydání: | 2007 |
Předmět: |
Applied Mathematics
Mathematical analysis Hilbert space Finite-rank operator Inverse problem Computer Science Applications Theoretical Computer Science Tikhonov regularization symbols.namesake Operator (computer programming) Rate of convergence Signal Processing Variational inequality symbols Applied mathematics Modes of convergence Mathematical Physics Mathematics |
Zdroj: | Inverse Problems. 23:987-1010 |
ISSN: | 1361-6420 0266-5611 |
DOI: | 10.1088/0266-5611/23/3/009 |
Popis: | There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems were developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita and co-workers presented a modification of the convergence rates result of Burger and Osher which turns out to be a complete generalization of the rates result of Engl and co-workers. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result from the previous ones is that the source condition is formulated as a variational inequality and not as an equation as previously. As examples, we present a phase retrieval problem and a specific inverse option pricing problem, both previously studied in the literature. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal. |
Databáze: | OpenAIRE |
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