Some application examples of minimization based formulations of inverse problems and their regularization

Abstract

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

“…Uniqueness of a minimizer of (32) will not necessarily hold; the sequence (x k ) k∈{1,…,k * } will therefore be defined by an arbitrary selection of minimizers of (32).…”

“…For further examples of such formulations, see., e.g., [16,32]. In particular we point to the variational formulation of EIT according to Kohn and Vogelius, see, e.g, [24], which will be extended to further diffusion/impendance identification problems in Sect.…”

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

“…For further examples of such formulations, see., e.g., [12,21]. In particular we point to the variational formulation according to Kohn and Vogelius, see, e.g, [20].…”

Analysis of an Inverse Source Problem with Correlation Data in Experimental Aeroacoustics

“…Moreover, one may ask if our source power reconstruction problem allows for a formulation that avoids the explicit evaluation of the forward operator (known as all-at-once formulations [Kal16,Kal18]). For the deterministic inverse source problem, such an approach has been investigated in [KKG18,GKK18,HK21]. Applying those ideas for the random inverse source problem with correlation data we obtain formally the following system of equations: find q and P such that F (q, P ) = 0 (physical model) , (8.1a)…”