A Two-stage Method for Inverse Medium Scattering

We present a novel numerical method to the time-harmonic inverse medium scattering problem of recovering the refractive index from near-field scattered data. The approach consists of two stages, one pruning step of detecting the scatterer support, and one resolution enhancing step with mixed regularization. The first step is strictly direct and of sampling type, and faithfully detects the scatterer support. The second step is an innovative application of nonsmooth mixed regularization, and it accurately resolves the scatterer sizes as well as intensities. The model is efficiently solved by a semi-smooth Newton-type method. Numerical results for two- and three-dimensional examples indicate that the approach is accurate, computationally efficient, and robust with respect to data noise.Comment: 18 pages, 5 figure

On A Generalization Of Regińska's Parameter Choice Rule And Its Numerical Realization In Large-scale Multi-parameter Tikhonov Regularization

A crucial problem concerning Tikhonov regularization is the proper choice of the regularization parameter. This paper deals with a generalization of a parameter choice rule due to Regińska (1996) [31], analyzed and algorithmically realized through a fast fixed-point method in Bazán (2008) [3], which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the single-parameter case, the algorithm does not require any information on the noise level. Further, combining projection over the Krylov subspace generated by the Golub-Kahan bidiagonalization (GKB) algorithm and the MFP method at each iteration, we derive a new algorithm for large-scale multi-parameter Tikhonov regularization problems. The performance of MFP when applied to well known discrete ill-posed problems is evaluated and compared with results obtained by the discrepancy principle. The results indicate that MFP is efficient and competitive. 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