Variational regularization of statistical inverse problems

Torsten Hohage (Universität Göttingen)
Wednesday, April 22, 2015 - 10:00am
WIAS, Erhard-Schmidt-Saal, Mohrenstraße 39, 10117 Berlin

We consider variational regularization methods for ill-posed inverse problems described by operator equations $F(x)=y$ in Banach spaces. One focus of this talk will be on data noise models: We will present a general framework which allows to treat many noise models and data fidelity terms in a unified setting, including Gaussian and Poisson processes, continuous and discrete models, and impulsive noise models. Rates of convergence are determined by abstract smoothness conditions called source conditions. In variational regularization theory these conditions are often formulated in the form of variational inequalities rather than range conditions for the functional calculus at $F'[x^{\dagger}]^*F'[x^{\dagger}]$ where $x^{\dagger}$ denotes the exact solution. Although this has a number of advantages from a theoretical perspective, there has been a lack of interpretations of variational source conditions for relevant problems. Here we will show for an inverse medium scattering problem that Sobolev smoothness of the contrast implies logarithmic variational source conditions and logarithmic rates of convergence for generalized Tikhonov regularization as the noise level tends to $0$. Our general results will be illustrated in the context of phase retrieval problems in coherent x-ray imaging, inverse scattering problems, and parameter identification problems in stochastic differential equations.}