A phaseless inverse scattering problem for the 3-D Helmholtz equation
Michael V. Klibanov
Inverse Problems & Imaging 2017, 11(2): 263-276 doi: 10.3934/ipi.2017013

An inverse scattering problem for the 3-D Helmholtz equation is considered. Only the modulus of the complex valued scattered wave field is assumed to be measured and the phase is not measured. This problem naturally arises in the lensless quality control of fabricated nanostructures. Uniqueness theorem is proved.

keywords: Lensless quality control of fabricated nanostructures high frequencies phaseless inverse scattering uniqueness
A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data
Michael V. Klibanov Dinh-Liem Nguyen Loc H. Nguyen Hui Liu
Inverse Problems & Imaging 2018, 12(2): 493-523 doi: 10.3934/ipi.2018021

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any a priori knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

keywords: Global convergence coefficient inverse problem inverse medium problem coefficient identification multi-frequency data experimental data single measurement

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