Characterizations of Variational Source Conditions, Converse Results, and Maxisets of Spectral Regularization Methods

We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of illposedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates of spectral regularization methods, which are slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. In many cases variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.