Parameter choice in Banach space regularization under variational inequalities

Abstract

Abstract. The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depends on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskiȋ principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.

Elastic-net regularization versusℓ1-regularization for linear inverse problems with quasi-sparse solutions

“…To overcome the difficulty, we shall use the variational inequalities (variational source conditions) instead. For more discussions about smoothness of solutions and their influences on convergence rates, we refer to [23,25] and [15,16,18] for further details.…”

“…Taking into account the estimates from [25], we can distinguish upper bounds of E η (x δ β * ,η , x † ) in Theorem 5.2 for β * = β T DP and β * = β SDP , which, however, yield the same convergence rate (5.9). To be more precise, we find for sufficiently small δ > 0 that η x δ β * ,η − x † ℓ 1 + 1 4 x δ β * ,η − x † 2 ℓ 2 ≤ C * (2η ϕ(δ) + Kψ(δ)) (5.14)…”

On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept

“…For more details concerning the consequences of variational inequalities and the role of the choice of the regularization parameter for obtaining convergence rates in regularization we refer, for example, to [19] and [1,4,5,12,15,21]. Making use of Gelfand triples it was shown in [2] that, for a wide range of applied inverse problems, the forward operators A are such that link conditions of the form (3.1) apply for all e (k) , k ∈ N. On the other hand, the paper [13] gives counterexamples where (3.1) fails for specific operators A, but alternative link conditions presented there can compensate this deficit.…”

Regularization Methods for Ill-Posed Problems

“…Along the lines of [107, Section 3.2], [109, Chapters 3 and 4], and [3,17,58,60,62,65], there are presented in the following some results on the regularization theory for that setting.…”