Dinh-Liem Nguyen

Dinh-Liem Nguyen, Michael V. Klibanov, Loc H. Nguyen et al.

This paper is concerned with the numerical solution to a 3D coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called "data propagation" procedure.

Dinh-Liem Nguyen, Michael V. Klibanov, Loc H. Nguyen et al.

We analyze in this paper the performance of a newly developed globally convergent numerical method for a coefficient inverse problem for the case of multi-frequency experimental backscatter data associated to a single incident wave. These data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. The challenges for the inverse problem under the consideration are not only from its high nonlinearity and severe ill-posedness but also from the facts that the amount of the measured data is minimal and that these raw data are contaminated by a significant amount of noise, due to a non-ideal experimental setup. This setup is motivated by our target application in detecting and identifying explosives. We show in this paper how the raw data can be preprocessed and successfully inverted using our inversion method. More precisely, we are able to reconstruct the dielectric constants and the locations of the scattering objects with a good accuracy, without using any advanced \emph{a priori} knowledge of their physical and geometrical properties.

Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen

This paper is concerned with a numerical method for a 3D coefficient inverse problem with phaseless scattering data. These are multi-frequency data generated by a single direction of the incident plane wave. Our numerical procedure consists of two stages. The first stage aims to reconstruct the (approximate) scattered field at the plane of measurements from its intensity. We present an algorithm for the reconstruction process and prove a uniqueness result of this reconstruction. After obtaining the approximate scattered field, we exploit a newly developed globally convergent numerical method to solve the coefficient inverse problem with the phased scattering data. The latter is the second stage of our algorithm. Numerical examples are presented to demonstrate the performance of our method. Finally, we present a numerical study which aims to show that, under a certain assumption, the solution of the scattering problem for the 3D scalar Helmholtz equation can be used to approximate the component of the electric field which was originally incident upon the medium.

Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen et al.

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 \textit{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.

Aleksandr E. Kolesov, Michael V. Klibanov, Loc H. Nguyen et al.

The recently developed globally convergent numerical method for an inverse medium problem for the Helmholtz equation is tested on experimental data. The data were originally collected in the time domain, whereas the method works in the frequency domain with the multi-frequency data. Due to a huge discrepancy between the collected and computationally simulated data, the straightforward Fourier transform of the experimental data does not work. Hence, it is necessary to develop a heuristic data preprocessing procedure. This procedure is described. The preprocessed data are used as the input for the inversion algorithm. Numerical results demonstrate good accuracy in the reconstruction of both refracive indices and locations of targets. Furthermore, the reconstruction errors for refractive indices of dielectric targets are significantly less than errors of a posteriori direct measurements.

Michael V. Klibanov, Aleksandr E. Kolesov, Dinh-Liem Nguyen

We present in this paper a novel numerical reconstruction method for solving a 3D coefficient inverse problem with scattering data generated by a single direction of the incident plane wave. This inverse problem is well-known to be a highly nonlinear and ill-posed problem. Therefore, optimization-based reconstruction methods for solving this problem would typically suffer from the local-minima trapping and require strong a priori information of the solution. To avoid these problems, in our numerical method, we aim to construct a cost functional with a globally strictly convex property, whose minimizer can provide a good approximation for the exact solution of the inverse problem. The key ingredients for the construction of such functional are an integro-differential formulation of the inverse problem and a Carleman weight function. Under a (partial) finite difference approximation, the global strict convexity is proven using the tool of Carleman estimates. The global convergence of the gradient projection method to the exact solution is proven as well. We demonstrate the efficiency of our reconstruction method via a numerical study of experimental backscatter data for buried objects.

Liliana Borcea, Dinh-Liem Nguyen

We study an inverse scattering problem for Maxwell's equations in terminating waveguides, where localized reflectors are to be imaged using a remote array of sensors. The array probes the waveguide with waves and measures the scattered returns. The mathematical formulation of the inverse scattering problem is based on the electromagnetic Lippmann-Schwinger integral equation and an explicit calculation of the Green tensor. The image formation is carried with reverse time migration and with $\ell_1$ optimization.