Abstract
We consider Tikhonov and sparsity-promoting regularization in Banach spaces for inverse scattering from penetrable anisotropic media. To this end, we equip an admissible set of material parameters with the $L^p$-topology and use Meyers' gradient estimate for solutions of elliptic equations to analyze the dependence of scattered fields and their Fréchet derivatives on the material parameter. This allows to show convergence of a non-linear Tikhonov regularization against a minimum-norm solution to the inverse problem, but also to set up sparsity-promoting versions of that regularization method. For both approaches, the discrepancy is defined via a $q$-Schatten norm or an $L^q$-norm with $1 < q < ∞$. Numerical reconstruction examples indicate the reconstruction quality of the method, as well as the qualitative dependence of the reconstructions on $q$.
Highlights
We consider direct and inverse scattering of time harmonic waves from a penetrable and anisotropic inhomogeneous medium with density described by a matrix-valued materialC contrast parameter Q ∈ d×dR div((Idd +Q)∇u) + k2u = 0 in d, d = 2, 3. (1)To this end, we set up weak solution theory for the scattering problem in Lebesgue spaces Lt with t ≥ 2 to be able to treat contrast functions in Lp with p < ∞ in some admissible set
We set up weak solution theory for the scattering problem in Lebesgue spaces Lt with t ≥ 2 to be able to treat contrast functions in Lp with p < ∞ in some admissible set
We show that different variants of Tikhonov regularization can be employed for this task and in particular suggest a sparsity-promoting variant of that technique
Summary
The inverse scattering problem we are interested in is to stably approximate the contrast function Qexa from noisy measurements of the far field pattern u∞, that is, from a noisy version Fmε eas such that Fexa − Fmε eas ≤ ε for some noise-level ε To this end, we show that different variants of Tikhonov regularization can be employed for this task and in particular suggest a sparsity-promoting variant of that technique. Following [11], we use Meyers’ gradient estimates for weak solutions to elliptic equations to obtain that gradients of weak solutions to (1) belong to Lt-spaces with t > 2 This in turn allows to prove various analytic properties for the solution to (1) as Lipschitz continuity or directional Gateaux differentiability that only require the contrast Q to be measured in some Lp-norm with 2 < p < ∞.
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