Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data
Year: 2012
Type: article
Abstract: In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations $\gdag = F(\udag)$ where $\gdag$ is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density $t\gdag$ where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\uı}-type parameter choice rule.
Source: Inverse Problems
Authors: Frank Werner, Thorsten Hohage
Institution: University of Göttingen
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G4818080411
OpenAlex ID: W1997446124