  We study the construction and updating of spectral preconditioners for
regularized Newton methods and their application to electromagnetic inverse
medium scattering problems. Moreover, we show how a Lepski\u{i}-type stopping
rule can be implemented efficiently for these methods. In numerical examples,
the proposed method compares favorably with other iterative regularization
method in terms of work-precision diagrams for exact data. For data perturbed
by random noise, the Lepski\u{i}-type stopping rule performs considerably
better than the commonly used discrepancy principle.
