Nonlinear Estimation for Linear Inverse Problems with Error in the Operator1 by Marc Hoffmann

The process Kδ is a blurred version of K , polluted by a Gaussian operator white noise Ḃ with a noise level δ > 0. The operator K acting on f is unknown and treated as a nuisance parameter. However, preliminary statistical inference about K is possible, with an accuracy governed by δ. Another equivalent approach is to consider that for experimental reasons we never have access to K in practice, but rather to Kδ . The error level δ can be linked to the accuracy of supplementary experiments; see Efromovich and Koltchinskii [11] and the examples below. In most interesting cases K−1 is not continuous and the estimation problem (1.1) is ill-posed (e.g., see Nussbaum and Pereverzev [16] and Engl, Hanke and Neubauer [12]). The statistical model is thus given by the observation (gε,Kδ). Asymptotics are taken as δ, ε→ 0 simultaneously. In probabilistic terms, observable quantities take the form 〈gε, k〉 := 〈Kf,k〉L2(Q) + ε〈Ẇ , k〉 ∀k ∈L2(Q)