Martin Burger
Institute for Computational and Applied Mathematics, University of Münster

"Uncertainty Quantification in the Variational Regularization of Inverse Problems"

Variational techniques in the regularization of inverse problems have evolved to become a standard tool in the field. In particular in image reconstruction, the use of nonquadratic regularization functionals (and possibly data fidelities) such as sparsity-promoting l1-minimization or edge-enhancing techniques based on total variation made enormous impact. Consequently novel questions with respect to the quantification of uncertainties were raised, questions of particular relevance being the relation to appropriate Bayesian prior and posterior models on the one hand and the estimation of errors caused by random noise in the data.

In this talk we will discuss several recent developments in these problems in important case of convex regularization functionals(log-concave priors in the Bayesian setup), based on the use of Bregman distances and other dual error measures. We discuss a quite general approach to estimate errors in the solutions of the variational regularization methods, which is based on duality techniques for convex optimization and can treat large (unbounded) noise. As a direct consequence we obtain estimates on the expected error for a setup with white noise in the data. Moreover, we discuss the characterization of the minimizer as a maximum a-posteriori probability (MAP) estimate for an appropriate model. For this sake we introduce the novel concept of weak MAP estimates and relate those to minimizers of a natural Bayes cost.

This talk is based on joint work with Tapio Helin (Helsinki), Felix Lucka (UCL) and Hanne Kekkonen (Warwick).