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arXiv · 1812.11327

Convergence rates for inverse Problems in Hilbert spaces: A Comparative Study

Gaurav Mittal·Ankik Kumar Giri
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Abstract

In this paper, we apply a new kind of smoothness concept, i.e. H\"older stability estimates for the determination of convergence rates of Tikhonov regularization for linear and non-linear inverse problems in Hilbert spaces. For linear inverse problems, we obtain the convergence rates without incorporating the classical concept of spectral theory and for non-linear inverse problems, we obtain the convergence rates without incorporating any additional non-linearity estimate. Further, we employ the smoothness concept of inhomogeneous variational inequalities to deduce the convergence rates for non-linear inverse problems. In addition to Tikhonov regularization, we also consider Lavrentiev's regularization method for non-linear inverse problems and determine its convergence rates by incorporating the H\"older stability estimates as well as inhomogeneous variational inequalities. And finally, we discuss the co-action between the variational inequalities and the H\"older stability estimates.

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Gaurav Mittal, Ankik Kumar Giri. 2018-12-29. Convergence rates for inverse Problems in Hilbert spaces: A Comparative Study. https://arxiv.org/abs/1812.11327

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