Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

Details Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

2012-04-07

arxiv.org/abs/1204.1669v1

Mathematics

Numerical Analysis

Scientific paper

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{\u\i}-type parameter choice rule.

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