Abstract. This paper is concerned with the classical inverse scattering problem to recover the refractive index of a medium given near or far field measurements of scattered time-harmonic acoustic waves. It contains the first rigorous proof of (logarithmic) rates of convergence for Tikhonov regularization under Sobolev smoothness assumptions for the refractive index. This is achieved by combining two lines of research, conditional stability estimates via geometrical optics solutions and variational regularization theory.

We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of ill-posedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates of spectral regularization methods, which are slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. In many cases variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.

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.

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