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

Abstract

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 ill-posedness 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.

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

“…If q 2 = ∞, that is f is in the largest space with smoothness s, then we also obtain q 3 = ∞. This is interesting because Besov spaces B s 2,∞ are known to be maximal sets for L 2 -regularization for certain problems (see [26]).…”

“…Our main tool for the derivation of variational source conditions will be the following generalization of Theorem 2.1 in [26]: Theorem 3.3. Let X and Y be Banach spaces and R a penalty term such that Assumption 3.1 is fulfilled.…”

“…Recently, the authors have developed a method for the verification of variational source conditions in Hilbert spaces under standard smoothness conditions. This method has been successfully applied to a number of interesting inverse problems, both linear and nonlinear [25,26,35,40].…”

Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

“…Variational source conditions are sufficient and often even necessary conditions for rates of convergence of Tikhonov regularization and other regularization methods ( [34,15,24]). In general such conditions have the following form: Let l :…”

The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

“…for some in the subdifferential ∂R(u) we make use of a variational source condition (cf., e.g., [2,7,8,[13][14][15]) at some solution u * ∈ B R of (1.1)…”