Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems

Abstract

This paper is concerned with the inverse problem to recover the scalar, complex-valued refractive index of a medium from measurements of scattered time-harmonic electromagnetic waves at a fixed frequency. The main results are two variational source conditions for near and far field data, which imply logarithmic rates of convergence of regularization methods, in particular Tikhonov regularization, as the noise level tends to 0. Moreover, these variational source conditions imply conditional stability estimates which improve and complement known stability estimates in the literature.

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

“…if T is compact we may choose ǫ = 0). We claim that ξ ′ (depending on α ′ and f † ) can be chosen such that (23) r…”

“…Concerning the first question, verifiable sufficient conditions for (3) have mainly been given via spectral source conditions so far, see Appendix A for more details. In [11,23] we have recently derived sufficient conditions for (3) in the form of bounds on Sobolev norms of f † for nonlinear inverse medium scattering problems. Here we formulate in Theorem 2.1 two criteria which capture the main strategy of the proofs in [11,23].…”

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

“…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].…”

Variational source condition for ill-posed backward nonlinear Maxwell’s equations

“…For abstract linear operators and p penalties with respect to certain bases, we refer to [2,5,7] and references therein. Recently, Hohage and Weidling [19,20] shown the validity of VSC for the Tikhonov regularization of inverse scattering problems. In particular, based on VSC, they obtained convergence with logarithmic-type rates for the corresponding regularized solution.…”

“…In particular, based on VSC, they obtained convergence with logarithmic-type rates for the corresponding regularized solution. For more details between VSC and classical source conditions, we refer the reader to [19][20][21]. See also [7] concerning recent results on VSC for elastic-net regularizations.…”