Tikhonov regularization in \(L^p\) applied to inverse medium scattering (Q2861871)

The authors analyze Tikhonov- and iterative shrunk Landweber regularization schemes for nonlinear inverse medium scattering problems. It is assumed that the contrast of the medium is supported within a small subdomain of a known search domain. The nonlinear Tikhonov functionals with sparsity-promoting penalty terms based on \(L^p\)-norms is minimized. Analytically, this is based on the scattering theory of the Helmholtz equation and on the crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, an iterated soft-shrinkage scheme combined with the differentiability of the forward operatorin \(L^p\) is used to approximate the minimizer of the Tikhonov functional. The quality of the obtained reconstructions is demonstrated using numerical examples.