Convergence rates and uncertainty quantification for inverse problems

My dissertation focuses on the convergence rates and uncertainty quantification for continuous linear inverse problems. The problem is studied from both deterministic and stochastic points of view. In particular, I considered regularisation and Bayesian inversion with large noise in infinite-dimensional settings. The first paper in my thesis investigates the convergence results for continuous Tikhonov regularisation in appropriate Sobolev spaces. The convergence rates are achieved by using microlocal analysis for pseudodifferential operators. In the second paper variational regularisation is studied using convex analysis. In this paper we define a new kind of approximated source condition for large noise and for the unknown solution to guarantee the convergence of the approximated solution in Bregman distance. The third paper approaches Gaussian inverse problems from the statistical perspective. In this article we study the posterior contraction rates and credible sets for Bayesian inverse problems. Also the frequentist confidence regions are examined. The analysis of the small noise limit in statistical inverse problems, also known as the theory of posterior consistency, has attracted a lot of interest in the last decade. Developing a comprehensive theory is important since posterior consistency justifies the use of the Bayesian approach the same way as convergence results justify the use of regularisation techniques.